UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Minimal (k)-groups, their structure and relevance to (G,x)-spaces Chan, Gin Hor 1971

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1971_A1 C34_3.pdf [ 5.59MB ]
Metadata
JSON: 831-1.0080485.json
JSON-LD: 831-1.0080485-ld.json
RDF/XML (Pretty): 831-1.0080485-rdf.xml
RDF/JSON: 831-1.0080485-rdf.json
Turtle: 831-1.0080485-turtle.txt
N-Triples: 831-1.0080485-rdf-ntriples.txt
Original Record: 831-1.0080485-source.json
Full Text
831-1.0080485-fulltext.txt
Citation
831-1.0080485.ris

Full Text

MINIMAL (k)-GROUPS , THEIR STRUCTURE AND RELEVANCE TO (G,x)-SPACES . by " GIN HOR CHAN B . S c , Nanyang Univ e r s i t y -Singapore, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS . We accept t h i s t h e s i s as conforming to the r e q u i r e d standard The" U n i v e r s i t y o f B r i t i s h Columb,!^. • X January 1 9 7 1 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a A B S T R A C T The problem of f i n d i n g a necessary and s u f f i c i e n t c o n d i t i o n f o r the t r i v i a l i t y of a (G , x )-space leads us to study and c l a s s i f y the p r o p e r t i e s of a minimal (k)-group G a c t i n g on N = {1,2,...,N} ( i . e . f o r each p a r t i t i o n P = {X^,X 2j...jX^} on N , there e x i s t s 1 | g e G wit h g ( X i ) = X ± f o r a l l i ) . In t h i s t h e s i s , xve co n s t r u c t and c l a s s i f y a l l the minimal .- (k)-groups of degree <_ Jk a.nd tabulate t:>e r e s u l t s . We al s o o b t a i n a l l the non - p r i m i t i v e (k)-grc ~ p s of degree <_ 4 k . We then apply our r e s u l t s to determine., which of the . (G, X.) -spaces are t r i v i a l . We found that f o r some of the (k)-groups, no appropriate character e x i s t s while f o r most of the remaining, the associated character has range { 1 , - 1 } . F i n a l l y , a t a b l e has been made to show the number of the appropriate characters on some (k)-groups. i . TABLE OF CONTENT PAGE CHAPTER I : I n t r o d u c t i o n t o ( G , x ) ~ s P a c e s a n d ( k ) - g r o u p s . 1 1 . ( G , x ) - s P a c e s • • 2 2. The t r i v i a l i t y o f a (G,x ) -space o v e r V . 5 3- ( k ) - g r o u p s ^ CHAPTER I I : ( k ) - g r o u p s o f degree n < 2k + 1 . . 7 1 . T r a n s i t i v i t y and p r i m i t i v i t y o f ( k ) - g r o u p s o f degree n .<• 2k + 1 . 7 \ 2\ ( k ) - g r o u p s o f degree n < 2k . . . . 10 '/ 3. ( k ) - g r o u p s o f degree n = 2k . . . 1J> 4 . ( k ) - g r o u p s o f degree n = 2k + 1 . 18 CHAPTER I I I : C o n s t r u c t i o n and examples o f m i n i m a l N ( k ) - g r o u p s . • • • • 25 1 . C o n s t r u c t i o n o f i m p r i m i t i v e . . m i n i m a l 'V .-j--^ ."v:r^ -.-2 . C o n s t r u c t i o n o f i n t r a n s i t i v e m i n i m a l ' ( k ) - g r o u p s w i t h n o f i x e d p o i n t , . . . . C H A P T E R I V : ( k ) - g r o u p s o f d e g r e e n , 2 k 4 - 2 j O V ' 3 k 1 . I n t r a n s i t i v e m i n i m a l ( k ) - g r o u p s o f d e g r e e n. , 2 k + 2 <_ n <: 3 k . ., .. , .; „ 2. T r a n s i t i v e ( k ) - g r o u p s o f d e g r e e : i , 2 n + 2 £ n £ 3 k - 2 3 - I m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p s o f d e g r e e n - 3 k . 4 . P r i m i t i v e ( k ) - g r o u p s o f d e g r e e n = 3 K - 1 o r 3 k C H A P l ' S R V : N o n - p r i m i t i v e (t)-groups o f d e g r e e n , 3 k < n < 4 k , I n t r a n s i t i v e m i n i m a l ( k ) - g r o u p s o f d e g r e e n , 3 k < n < 4 k . . . . . . . . . i i i . 2 . Theorems on i m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p s 8 1 3 I m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p s o f degree n , ) k < n < 4k 84 CHAPTER V I : The ( k ) - c h a r a c t e r s o n m i n i m a l ( k ) - g r o u p s and i t s a p p l i c a t i o n t o t h e t r i v i a l i t y o f a ( G , X ) - s p a c e 91 1 . ( k ) - c h a r a c t e r s on m i n i m a l ( k ) - g r o u p s o f t ype S k + 1 and A^+2 92 2. ( k ) - c h a r a c t e r s on m i n i m a l ( k ) - g r o u p s o f t y p e H k 92 3. ( k ) - c h a r a c t e r s on m i n i m a l ( k ) - g r o u p s o f t y p e S k \ A } . 94 4 . ( k ) - c h a r a c t e r s on m i n i m a l ( k ) - g r o u p s o f t y p e Q k o r 95 5. ( k ) - c h a r a c t e r s on i n t r a n s i t i v e m i n i m a l ( k ) - g r o u p s 96 \ -BIBLIOGRAPHY 103 ACKNOWLEDGEMENTS I am i n d e b t e d t o my s u p e r v i s o r . P r o f e s s o r JR. Westwick, f o r h i s generous and v a l u a b l e a s s i s t a n c e d u r i n g the r e s e a r c h and w r i t i n g o f t h i s t h e s i s . I would a l s o l i k e to thank P r o f e s s o r 3. Chang and P r o f e s s o r A. Mowshowitz f o r t h e i r c r i t i c i s m s and r e a d i n g o f t h i s t h e s i s . I am g r a t e f u l t o The U n i v e r s i t y o f B r i t i s h Columbia and N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r t h e i r f i n a n c i a l a s s i s t a n c e . L a s t , b u t not l e a s t , I w i s h to thank Eve Hamilion f o r t y p i n g t h i s t h e s i s . C h a p t e r I I n t r o d u c t i o n t o ( G , x ) - s p a c e s a n d ( k ) - g r o u p s A s p a c e o f s y m m e t r y d a s s e s o f t e n s o r s , o r m o r e s p e c i f i c a l l y , a ( G , x ) - s p a c e o v e r a v i c t o r s p a c e V h a s b e e n d e f i n e d b y M a r c u s a n d M i n e i n £ 2 j , a n d m o r e g e n e r a l l y b y S i n g h i n [ 3 ] - T h e d e f i n i t i o n d e p e n d s o n a c h o i c e o f a p e r m u t a t i o n g r o u p G a n d a l i n e a r c h a r a c t e r ;.' o f G . F o r s o m e c h o i c e s t h e r e s u l t i n g ' s p a c e i s t r i v i a l ( e . g . t h e G r a s s m a r m s p a c e w h e n t h e d i m e n s i o n o f V i s ' l e s s t h a n t h e d e g r e e o f G ).. I n t h i s t h e s i s we a t t e m p t - t o -r s i n e . j u s t w h e n t h i s i s t h e c a s e . T h i s l e a n s u s zo .. "\6efining a s p e c i a l ' t y p e o f p e r m u t a t i o n g r o u p •( f k . ) - g r o u p s ) , • a n d • st'he m a j o r p a r t o f t h e t h e s i s i s i n v o l v e d w i t / , c h a r a c t e r Izl'xg t h o s e g r o u p s . T h e g r o u p t h e o r e t i c a s p e c t o f t h e t h e s i s do-.;.-n o t t a k e i n t o a c c o u n t - , t h e p o s s i b i l i t y o r i m p o s s i b i l i t y o : a n a p p r o p r i a t e c h a r a c t e r . S o m e o f t h e ( k ) - g r o u p s i n f a c e u o n o t g i v e r i s e t o a t r i v i a l ( G , x ) - s P a c e b e c a u s e n c n o n - t r i v i a l c h a r a c t e r i s a v a i l a b l e . I n t h e l a s t c h a p t e r we s h o w t h a t t h e r e e x i s t n o n - t r i v i a l c h a r a c t e r s f o r s o m e o f t h e ( k ) - g r o u p s a n d t a b u l a t e o u r r e s u l t s a t t h e e n d . *' I n t h i s c h a p t e r we f i r s t d e s c r i b e t h e i d e a o f ( G , x ) - s p a c e s , s t a t e m e n t o f t h e p r o b l e m o f t h i s t h e s i s . 2 . a n d i n t r o d u c e t h e c o n c e p t o f ( k ) - g r o u p . T h r o u g h o u t t h i s t h e s i s , we l e t F = a n a r b i t r a r y f i e l d . Q f c h a r a c t e r i s t i c + 2 . V = a v e c t o r s p a c e o v e r F o f d i m e n s i o n k . N = { 1 , 2 , . . . , n ) G = a p e r m u t a t i o n g r o u p o n N . S y m ( N ) = S n = t h e s y m m e t r i c g r o u p o n N . A l t ( N ) = A n = t h e a l t e r n a t i n g g r o u p o n N . X = a l i n e a r c h a r a c t e r o n G o v e r F . § 1 , ( G , x ) - s p a c e s D e f i n i t i o n 1 . 1 . L e t V n = V x . . . x V ( n t i m e s ) . A f u n c t i o n f • V1 -• W s w h e r e ¥ i s a v e c t o r s p a c e o v e r F , i s s a i d t o b e m u l t i l i n e a r i f f o r e v e r y x = ( x - ^ X g , ,. . , x ) , y •= > y i a y 2 * • • - > y n ) • > z = ( z i > z 2 » * • *  ,zn>  e ^  a n d a »  9 € F s u c h t h a t f o r s o m e i , l < _ i _ < n , ( 2 ) x ± = a y j _ + p z . t h e n X' f ( x ) = a f ( y ) + pf(z) . I f n = 2 , f i s c a l l e d a b i l i n e a r m a p . D e f i n i t i o n 1 . 2 . A f u n c t i o n f = V 1 1 - ¥ i s c a l l e d a ( G , X) - m a p i f f o r e v e r y g e G a n d x = ( x ^ x ^ , . . . >x^) e V 1 1 , f ( x g ( l ) , . . . , x g ( n ) ) = x ( g ) - f ( x 1 , x 2 , . . . x n ) . D e f i n i t i o n 1 . 3 - L e t V be' a v e c t o r s p a c e , a p a i r ( U , ( b ) i s c a l l e d a ( G , \ ) - s p a c e o v e r V i f (1) U i s a v e c t o r s p a c e o v e r F (2) (]) : V 1 1 - U i s a m u l t i l i n e a r ( G , x ) - m a p ( 3 ) ' F o r a n y v e c t o r s p a c e W o v e r F, a n d a n y . m u l t i l i n e a r ( G , x ) - m a p f : V a W , t h e r e e x i s t s a u n i q u e h o m o m o r p h i s m f : U - w s u c h t h a t f (j) = f . C o n d i t i o n ( 3 ) i n D e f i n i t i o n 1 . 3 i s t h e u n i v e r s a l m a p p i n g p r o p e r t y f o r m u l t i l i n e a r ( G , x ) - m a p s . T h e f o l l o w i n g T h e o r e m a s s e r t s t h e e x i s t e n c e o f ( G , x ) - s p a c e o v e r V f o r a l l V . T h e o r e m 1.k. [ 3 ] F o r e a c h G , X; a n d V , t h e r e e x i s t s a ( G , x ) - s p a c e (U,(J)) o v e r V , a n d U i s u n i q u e u p t o i s o m o r p h i s m . §2. T h e t r i v i a l i t y o f a ( G , x ) - s p a c e o v e r V . T h e m a i n t h e m e o f t h e p r e s e n t t h e s i s i s t o f i n d o u t w h e n a ( G , x ) - s p a c e o v e r V i s t r i v i a l . F o r a G r a s s m a n s p a c e , i t i s t r i v i a l i f a n d o n l y i f d i m V = k < n T h e T e n s o r s p a c e s a n d S y m m e t r i c s p a c e s a r e a l w a y s n o n -t r i v i a l . H e n c e t h e t r i v i a l i t y o f a ( G , X ) - s p a c e o v e r V d e p e n d s o n t h e v a l u e s o f k a n d n , t h e g r o u p G a n d t h e l i n e a r c h a r a c t e r x . 4. L e t (U,(j)) b e a ( G , X ) - s p a c e o v e r V . We d e n o t e a n e l e m e n t (J) ( x - ^ . . . , x n ) i n U b y x - ^ A . . . A x n . L e t B = { y 1 } y 2 3 • • < j y k ) b e a b a s i s o f V . T h e n i t i s n o t d i f f i c u l t t o c h e c k t h a t t h e s e t o f a l l y o A y A . . . A y , {y , y . . . , y } c B , s p a n s u . "1 s2 s n s l s2 s n ~~ S i n g h [3] g a v e a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r y q A y A . . . A y „ = 0 . H e d e f i n e d ( y , . . . , y ) t o s l s2 s n s l s n b e a ( G , X ) - e l e m e n t i f t h e r e e x i s t s ! g e G w i t h X ( S ) 4 1 s u c h t h a t ( y „ , . . . , y g ) = ( y > - • >ys )> °1 n g ( l ) g ( n ) a n d p r o v e d . L e m m a 1.6. A n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r y A y A . . . A y = 0 i s t h a t ( y , y , . . . , y n ) i s a s l s2 n .. s l s2 ° n ( G , X ) - e l e m e n t . F r o m L e m m a 1.6 i t f o l l o w s t h a t a n e c e s s a r y c o n d i t i o n f o r a ( G , X ) - s p a c e o v e r V t o b e t r i v i a l i s t h a t d i m V = k < n a n d x i s a n o n - t r i v i a l c h a r a c t e r . F o r d i m V = 1 , i t i s c l e a r t h a t e v e r y n o n -t r i v i a l l i n e a r c h a r a c t e r x ' ' G -• F w i l l m a k e a n y ( G , X ) - s p a c e ,(U,(|)) o v e r V t r i v i a l . H e n c e , i n w h a t f o l l o w s , we m a y a s s u m e k > 2 a n d k < n . §3. ( k ) - g r o u p s L e t B = { y ^ , y 2 , . . . , y ^ } b e a b a s i s o f V 5-F o r e a c h y = ( y , . . . , y ) e V 1 , w h e r e {y , . . . , y } c B , s l n s l . n ~ we d e f i n e N_. = { j | y = y.. } f o r a l l i = l , 2 , . . . , k . T h e n { N - ^ N g , . . . , N k ) f o r m s a p a r t i t i o n o f N , a n d y c A . . . A y „ = 0 i f a n d o n l y i f t h e r e e x i s t s g e G s l s n w i t h g ( N ± ) = N ± , i = 1,2,...,k a n d x ( g ) + 1 • W e a r e t h u s l e d t o t h e f o l l o w i n g d e f i n i t i o n s . D e f i n i t i o n 1.7. G i v e n a p a r t i t i o n P = { N , ,N,1,. . . , K ] o f N , a n . e l e m e n t 1 | g e S f l i s s a i d t o b e a P - e l e m e n t i f g ( N ^ ) = f o r a l l i . We s i m p l y w r i t e g ( P ) = P . D e f i n i t i o n 1.8. A p e r m u t a t i o n g r o u p G o n N i s s a i d t o b e a ( k ) - g r o u p i f f o r e v e r y p a r t i t i o n P = { N n , N 0 , . . , N , } t h e r e e x i s t s a P - e l e m e n t i n G . I f n <_ k ' , t h e n c l e a r l y n o ( k ) - g r o u p e x i s t s . F u r t h e r m o r e , t h e n u m b e r o f p o s s i b l e t y p e s o f k - g r o u p s i n c r e a s e s a s n i n c r e a s e s . W h e n n <_ 3k , we a r e a b l e t o d e t e r m i n e a l l y t h e m i n i m a l ( k ) - g r o u p s . T h i s i s d o n e i n C h a p t e r I I a n d I V , a n d s u m m a r i z e d i n a t a b l e a t t h e e n d o f C h a p t e r I V . W h e n n <_ 4k we a r e a b l e t c o b t a i n a l l t h e n o n - p r i m i t i v e ( k ) - g r o u p s . A,-.- f i n d i t c o n v e n i e n t t o i n s e r t h e r e s o m e e l e m e n t a r y p r o p e r t i e s o f ( k ) - g r o u p s . P r o p e r t i e s : ^ (1) A - ( k ) - g r o u p i n S n i s a ( k ) - g r o u p i n S m i n t h e e m b e d d i n g o f S n i n t o S m f o r n <_ m . (2) A ( k+1) - g r o u p i n S n i s a l s o a ( k ) - g r o u p i n S n 6. (3) ' E v e r y ( k ) - g r o u p c o n t a i n s a m i n i m a l ( k ) - g r o u p ( n o t n e c e s s a r i l y u n i q u e ) . . (4) N o ( k ) - g r o u p i s c y c l i c . (5) L e t G b e a m i n i m a l ( k ) - g r o u p . T h e n t h e r e e x i s t s n o n o r m a l s u b g r o u p - H o f G s u c h t h a t G / H i s c y c l i c a n d o f o r d e r p a , a > 1 a n d p i s a p r i m e . P r o o f ; ( i ) _ ( 4 ) a r e t r i v i a l . _We p r o v e (5) a s f o l l o w s S u p p o s e t h e c o n t r a r y . L e t H b e a n o r m a l s u b g r o u p o f G d e s c r i b e d a s i n (5)- L e t G / H = < g H > a n d (j) : G -• G / H b e t h e n a t u r a l h o m o m o r p h i s m . We s h o w t h a t t h e i n v e r s e i m a g e H 1 - (JT ( < g p H > ) I s a ( k ) - g r o u p . L e t P b e a n y p a r t i t i o n o n N a n d g ^ a P - e l e m e n t i n ' G . I f k Hj_"'i t h e n g ^ = g S h f o r s o m e h e K a n d s o m e s w i t h ( s . p ) = 1 . T h e r e f o r e si t p a m = 1 f o r s o m e t a n d m , a n d h e n c e g , ^ = ( g h ) = ( g " h 1 ) p p J- -1-= (gh-^) = g h ^ i s i n f o r s o m e h - ^ h g e H . p p I S i n c e g ^ -' =j= 1 i t i s a P - e l e m e n t i n . T h i s c o m p l e t e s t h e p r o o f . X 7 . Chapter I I On the (k)-groups o f degree n £ 2k + 1 The aim of t h i s chapter i s to f i n d a l l the (k)-groups c f degree n <_ 2k + 1 . For the d e f i n i t i o n s o f the terms r e l a t i n g to permutation groups used i n t h i s chapter one may r e f e r to Wielandt [5 ] . § 1 . T r a n s i t i v i t y and p r i m i t i v i t y o f (k)-groups o f degree n ± 2 k + 1 . Let G ,be a permutation group on N and_ l e t £ N . We d e f i n e G N = {g € G I g(a) = a , V a e N±) ; i N, G = { ( g l ^ ) I g(N^) = N x and g e G] N l When no c o n f u s i o n a r i s e s , G may be c o n s i d e r e d as a subgroup o f Sym(N) which leave a l l the elements i n N - f i x e d , Theorem 2 .1 . Let n <_ 2k + 1 and l e t G be a minimal ( k ) -group o f degree n . Then G ' i s " t r a n s i t i v e . Proof: Suppose G i s not t r a n s i t i v e and l e t N,,..,N X • 1 s . be the o r b i t s o f G . Then there i s a t l e a s t one o r b i t , say N x , such t h a t 2 < IN-J = t <_ k . Let N-^  ( x 1 , . . , x t ) . 8. We s h a l l s h o w t h a t i s a ( k ) - g r o u p o n N - a n d h e n c e i s a ( k ) - g r o u p o n N . F o r , g i v e n a n y p a r t i t i o n p = { X - j ^ X g , . . . . ,X^} o n N - , we f o r m a p a r t i t i o n p x = [ X 1 U [ x 1 } , . . . , X t u ( x t ) , X t + 1 , . . . , X k ) o n N . B y t h e a s s u m p t i o n o n G , t h e r e i s a P ^ - e l e m e n t g i n G . C l e a r l y g € G ^ , s i n c e e a c h o f t h e s e t s i n t h e p a r t i t i o n P.. i n t e r s e c t s i n a s i n g l e t o n o r n o t a t a l l . H e n c e G ^ i s a ( k ) - g r o u p p r o p e r l y c o n t a i n e d i n G . T h i s c o n t r a d i c t s t h e m i n i m a l i t y o f G . T h e p r o o f o f t h e T h e o r e m 2 . 1 y i e l d s t h e f o l l o w i n g t h e o r e m . T h e o r e m 2 . 2 . L e t G b e a ( k ) - g r o u p o f d e g r e e ' n <_ 2 k + 1 . T h e n t h e r e e x i s t s N-^ c N s u c h t h a t >. k + 1 a n d G ^ i s a ( k ) - g r o u p w h i c h i s t r a n s i t i v e o n N - . \ P r o o f : L e t H b e m i n i m a l ( k ) - g r o u p c o n t a i n e d i n G . L e t N-, = {x e N | h ( x ) = x , V h e H} . T h e n H c G . T i N l a n d i s t r a n s i t i v e , o n N - b y T h e o r e m 2 . 1 . S i n c e I N - N - J i s t h e d e g r e e o f H a n d H i s a ( k ) - g r o u p , | N - N x | > k + 1 . T h e o r e m 2 . 3 . I f n <_-2k + 1 a n d n + 2 k , t h e n e v e r y t r a n s i t i v e ( k ) - g r o u p G o f d e g r e e n i s p r i m i t i v e . 9-P r o o f : We f i r s t c o n s i d e r t h e c a s e w h e n n = 2 k + 1 . F o r k = 2 o r 3 , n = 2 k + 1 = 5 o r 7 , a n d G i s o f p r i m e d e g r e e , a n d h e n c e i s p r i m i t i v e . S o we s u p p o s e k > 4 . A s s u m e G i s n o t p r i m i t i v e . L e t { N - ^ N g s • • , N } b e a c o m p l e t e b l o c k s y s t e m o f G . S i n c e n = s j N - J = s t a n d n i s o d d , s > 3 a n d t > 3 . - L e t N . = { a . 1 3 a . Q , . . , a . .3* i - 1 , 2 , . . . , s . We h a v e t w o c a s e s : C a s e 1 . • s <_ t . We c o n s t r u c t a p a r t i t i o n P = { X o , X 1 , . . . , X t 3 o n N a s f o l l o w s : X o = { a l , l ' a 2 , 2 > " ^ a s , s } X i = t a l , i ' a 2 , i ' * ' - > a s , i } " t a i , i } 1 = 1 > 2 > - - > s • X s + j = ^ a l , s + ^ a 2 , s + t r ' ' - , a s , s + j 3 j = l , 2 , . . . , t - s . S i n c e t h e p a r t i t i o n P o n N h a s l e n g t h l e s s t h a n o r e q u a l t o t + 1 <_ ( k - 1 ) + 1 = k , t h e r e e x i s t s a P - e l e m e n t g i n G . We c l a i m t h a t g ( N . ) 4 N - f o r a H 1 > J • C l e a r l y , b y t h e c o n s t r u c t i o n o f P , g ( N ^ ) 4 f o r a l l i . A l s o i f g ( N . ) = N . f o r s o m e j 4 i > t h e n <t> = g((|)) = g (N^ nx^ ) = g(N±) n g(xi) = N j n X j + <|) , a c o n t r a d i c t i o n . B u t t h i s i s i m p o s s i b l e b e c a u s e { N - ^ N g . . . . ,N g 3 ] i s a c o m p l e t e b l o c k s y s t e m o f G . H e n c e G i s p r i m i t i v e . C a s e 2 . ', t < s . P u t s = m t + r , 0 <_ r < t . I f r = 0 , t h e n s = m t . We d i v i d e t h e c o m p l e t e b l o c k s y s t e m [ N , , N Q , . . . , N } i n t o m p a r t s , e a c h c o n s i s t i n g o f e x a c t l y t b l o c k s . T h e n , b y u s i n g t h e s a m e m e t h o d o f p a r t i t i o n a s i n c a s e 1 o n e a c h p a r t , we g e t t h e n e c e s s a r y c o n t r a d i c t i o n . 1 0 . I f r 4 0 , without l o s s of g e n e r a l i t y , we may assume m = 1 . Consider a p a r t i t i o n P = [X-^ . . . X 2 t ) on N d e f i n e d as f o l l o w s : X i = t a i , i ' a i + l , i ' - - - ' a i + t , i } 1 = i*2*'-'** • x t + j = ^i,y&2,y - • • >&s,^ " x j J = i , 2 , . . . , t . S i n c e n = 2 k " + 1 and each X . has a t l e a s t two elements l f o r a l l . i = l , 2 , . . . , 2 t , the p a r t i t i o n P i s of l e n g t h <_ k . B y assumption o f G there e x i s t s a P-element g e G . T h i s i s i m p o s s i b l e because g(N.)>4 N. f o r a l l 1 , j = l , 2 , . . , s . i J Hence G i s p r i m i t i v e . Now, we prove the case when "n < 2 k . Assume the c o n t r a r y and l e t N^ , ... , N g be d e f i n e d as above. We c o n s t r u c t a p a r t i t i o n P = ' ' ' 3^s+t 1 ^ o n ^ a s f o l l o w s : X i = * a i , l ) 1 = 1 , 2 , . . . , s . s+i = £ a i , i + Y a 2 , i + l ' - - ^ A s , i + l 3 1 = 1 * 2 * - - - ^ - 1 . • Since s _> 2 and t > 2 , we have s + t - l < _ | i + l = ^ + l < _ k , and hence the p a r t i t i o n P i s ' o f l e n g t h <_ k . Then there i s a P-element g e G w i t h g ( a 1 1 ) = &± ^ arid g(H^) 4 N i , i = l , 2 , . . . , s . But t h i s i n t u r n i m p l i e s g = 1 , a c o n t r a d i c t i o n . T h i s completes the p r o o f . § 2 . (k)-groups of degree n < 2 k L e t G^ and Gg be two permutation groups contained i n S n . We say t h a t G-^  and Gg are of the . 1 1 . s a m e t y p e i f t h e r e i s a g e S n s u c h t h a t g ~ ^ G ^ g = G 2 , i . e . t h e y a r e c o n j u g a t e . A s e x a m p l e s o f ( k ) - g r o u p s , we h a v e t h e g r o u p S ^ - L a n d t h e a l t e r n a t i n g g r o u p A k + 2 . T o s h o w i s a ( k ) - g r o u p , l e t P = { X - ^ X g , . . . ,X^} b e a p a r t i t i o n o f N o f l e n g t h k . T h e n a t l e a s t o n e o f X . , s a y X , c o n t a i n s m o r e t h a n o n e e l e m e n t . L e t a , p e X ^ . T h e n (af3) i s a P - e l e m e n t i n S^-f-1 a n d h e n c e i s a ( k ) - g r o u p . M o r e o v e r , i t i s n o t d i f f i c u l t t o s e e t h a t i s m i n i m a l . S i m i l a r l y , we c a n s h o w t h a t ^ - + 2 ^ s a m i n i m a l (k") - g r o u p . i n t h i s s e c t i o n , we s h a l l s h o w t h a t e v e r y ( k ) -g r o u p o f d e g r e e n < 2 k c o n t a i n s a s u b g r o u p o f t y p e o r 2 ( i f n >_ k + 2 ) L e m m a 2 . 4 . L e t G b e a ( k ) - g r o u p i n S . L e t A £ N , A ' = N - A . I f S y m ( A ) 0 G = {1 ) , t h e n G A ' i s a ( k - 1 ) - g r o u p a c t i n g o n A ' . P r o o f : L e t P = { X - ^ X g , . . / ^ X ^ - ^ ) t>e a n y p a r t i t i o n o n A 7 . T h e n P 1 = { A , X 1 , . . , X k _ 1 } i s a p a r t i t i o n o n N o f l e n g t h < k , a n d h e n c e t h e r e e x i s t s a P ^ - e l e m e n t g e G . We n o t e t h a t t h e r e s t r i c t i o n g | A ' 4 1 • F o r i f n o t , 1 4 g e S y m ( A ) D G = {1} , a c o n t r a d i c t i o n . H e n c e G i s a ( k - 1 ) - g r o u p o n A ' . -T h e f o l l o w i n g t h e o r e m i s d u e t o P r o f e s s o r W e s t w i c k . T h e p r o o f g i v e n h e r e , h o w e v e r , i s d i f f e r e n t f r o m h i s . 1 2 . Theorem 2 . 5 . I f n < 2 k , then every (k)-group i n S n c o n t a i n s a subgroup of type s k + ± o r ^ + 2 ( i f n >_ k + 2 ) . P r o o f : We prove t h i s theorem by i n d u c t i o n on n , the degree o f the (k)-group G . For n = 3 »• we must -have k = 2 and c l e a r l y G t c o n t a i n s S-, i n t h i s case. For n = 4 , we must have 3 k = 3 and again i t i s c l e a r t h a t G c o n t a i n s .• Suppose the theorem i s true f o r a l l groups of degree s where s < n , and f o r a l l . k such t h a t k < s < 2 k Let G be a (k)-group i n S. and l e t H be a minimal (k)-group contained i n G . I f H f i x e s some p o i n t s i n N , then H can be c o n s i d e r e d as a (k)-group o f degree l e s s than n and the i n d u c t i o n h y p othesis a p p l i e d to H p r o v i d e s us with a group of type S k + 1 or . I f H has no f i x e d p o i n t , then by Theorem 2 . 1 H i s t r a n s i t i v e on N . I f Sym (N) =}= H , theri (ap) £ H f o r some a , (3 e N . Let A = N - . Then H A i s a ( k - 1 ) - g r o u p o f degree .<_ n - 2 by Lemma 2 . 4 . Hence 7 by the i n d u c t i o n h y p o t h e s i s A 1 H c o n t a i n s a subgroup o f type S^ or , where k••_> .3 . T h i s i m p l i e s t h a t G c o n t a i n s the alternating group A i t ( N ^ ) on some s e t c A where |Nx| = k . Since H i s t r a n s i t i v e and n < 2k , there i s g e G such t h a t g(N 1) n N ± 4 ^ or , and xso G c o n t a i n s A l t ( N 2 ) f o r some N 2 ? N . I f j NpI > k + 1 , then G c o n t a i n s a subgroup of type A ^ 0 and we are done. Otherwise |N2| = k + 1 . I f n - k -i- 1 , then N 2 = N . Since A l t (N 2) i s not a ( k )-group we must have Sym (N) c H , a c o n t r a d i c t i o n . T herefore n > k + 1 , and hence by the same argument as above we can f i n d a subset N-, o f N such that + N„ 3 3 2 and A l t ( N ^ ) c G . T h e r e f o r e G c o n t a i n s a subgroup o f type A k + 2 . This completes the pr o o f . C c r o l l o r y 2 . 6 . I f n < 2 k , then every t r a n s i t i v e (k)-group o f degree n c o n t a i n s the a l t e r n a t i n g group A n . §3- (k)-groups of degree n = 2 k Let and N 2 be two subsets o f N with N 1 0 N 2 = (j) , U N 2 = N and IN-J = |N2| = k . Let • g 1 be a permutation on N which interchanges the elements i n N with those i n N 2 .. Let H ^ N ^ N g ) ^ < g 1 , Sym(N 1) , Sym(N 2) > . (In the sequ e l we w i l l c a l l such a group one of type H k) . Then Lemma 2.J. H^N-^Ng) i s a minimal (k)-group of degree n = 2 k . Proof: Since Sym (N-jJ and Sym (N 2) are k - f o l d t r a n s i t i v e and g.^  e H^N^Ng) , H ^ N p N g ) c o n t a i n s a l l the permutations which interchange the elements i n 1 4 . w i t h t h o s e i n N g . L e t P = [ X - ^ X g , . . . , X k ) b e a p a r t i t i o n o n N . I f X . n N . h a s m o r e t h a n o n e e l e m e n t f o r s o m e i a n d j , t h e n . ( a B ) i s a P - e l e m e n t i n H ^ N - ^ N g ) f o r s o m e a , B e X i n N j . O t h e r w i s e X ^ fl N . h a s o n l y o n e e l e m e n t f o r a l l i = l , 2 , . . . , k a n d j = 1 , 2 . T h i s i n d u c e s a o n e - t o - o n e c o r r e s p o n d e n c e h b e t w e e n e l e m e n t s i n N ^ a n d t h o s e i n N 2 s u c h t h a t h ( X i n N 1 ) = X i n N 2 a n d h ( X i n N 2 ) = X± n N x f o r a l l " i . C l e a r l y h i s a P - e l e m e n t i n H k ( N 1 , N 2 ) . H e n c e H ^ N ^ N g ) i s a ( k ) - g r o u p o n N o f d e g r e e 2 k . M o r e o v e r , s i n c e e v e r y ( k ) - g r o u p i n H ^ N ^ N g ) c o n t a i n s a l l t h e t r a n s p o s i t i o n s a n d a l l t h e p e r m u t a t i o n s o f o r d e r 2 w h i c h i n t e r c h a n g e t h e e l e m e n t s i n N ^ w i t h t h o s e i n N 2 , . H ^ N p N g ) ! i s m i n i m a l . L e m m a 2 . 8 . L e t G b e a m i n i m a l ( 2 ) - g r o u p o f d e g r e e 4 . T h e n G = A ^ o r G = H 2 ( N 1 , N 2 ) f o r s o m e TS. , N 2 w i t h N ] _ U N 2 = N a n d | N j = |Ng| = 2 . P r o o f : B y T h e o r e m 2 . 1 G i s t r a n s i t i v e . S i n c e t h e p r i m i t i v e g r o u p o f d e g r e e 4 m u s t c o n t a i n A ^ , we a r e d o n e i f G i s p r i m i t i v e . Now s u p p o s e G i s n o t p r i m i t i v e . L e t {N-j-jNg} b e a c o m p l e t e b l o c k s y s t e m o f G . S i n c e t h e m a x i m a l s u b g r o u p o f ' S ^ w i t h c o m p l e t e b l o c k s y s t e m • { N ^ N g } i s HgtN^Ngj , a n d i t i s a ''(2)-group, we h a v e G = H p ( N n , N p ) . . . - . . . . 15. L e m m a 2 . 9 . L e t G b e a p r i m i t i v e g r o u p o f d e g r e e 6 . I f . A l t ( N ) 4: G , t h e n G i s o f t y p e < (126) (3.5^) ' , ( 1 2 3 4 5 ) , ( 2 5 ) ( 3 * 0 > ( S e e p a g e 2 1 6 o f [ 1 ] ) , . T h e o r e m 2 . 1 0 . L e t G b e a t r a n s i t i v e ( k ) - g r o u p o f d e g r e e n = 2 k . T h e n G c o n t a i n s e i t h e r A n o r H k ( N 1 } N 2 ) w h e r e N ± U N 2 = N a n d I N - J = | N 2 1 = k . P r o o f : We u s e i n d u c t i o n o n k t o p r o v e t h i s t h e o r e m . F o r k = 2 , i t i s t r u e b y L e m m a 2 . 8 . A s s u m e t h e t h e o r e m i s t r u e f o r k - 1 w h e r e k > 2 . L e t G b e a t r a n s i t i v e ( k ) - g r o u p i n S n w h e r e n = 2 k . S u p p o s e G 4 s n • L e t a , B e N b e s u c h t h a t ( a B ) £ G a n d l e , t A = N - { a , 6 } . T h e n G A i s a ( k - 1 ) - g r o u p , o n A . L e t H b e a m i n i m a l ( k - l ) - g r o u p c o n t a i n e d i n G A . I f H f i x e s s o m e p o i n t i n A t h e n H c o n t a i n s a s u b g r o u p o f t y p e S ^ o r A ^ b y T h e o r e m 2 . 5 . I f H f i x e s n o p o i n t i n A , t h e n VH i s t r a n s i t i v e a n d h e n c e , b y t h e i n d u c t i o n h y p o t h e s i s - , - H c o n t a i n s a s u b g r o u p o f t y p e A n - 2 o r H k - 1 • C a s e 1 . G A c o n t a i n s a s u b g r o u p o f t y p e A k + 1 . T h i s i m p l i e s t h a t G c o n t a i n s a s u b g r o u p o f t y p e A k + X • S i n c e n = 2 k a n d G i s t r a n s i t i v e , we h a v e G :o A . • — n C a s e 2 . G A c o n t a i n s a s u b g r o u p o f t y p e A ^ . . T h i s i m p l i e s G 3. A l t (A-^ f o r s o m e A x c A a n d JA-J = k . 16... I f t h e r e e x i s t s a g e G s u c h t h a t g ( A 1 ) fl A-j_ =j= (j) o r A-^  , t h e n G c o n t a i n s t h e a l t e r n a t i n g g r o u p o n A U g ( A 1 ) , a n d t h i s r e d u c e s t o c a s e 1 . O t h e r w i s e , s u p p o s e g ( A x ) n A-j_ = o r A x f o r a l l g € G ; i . e . , {A^N-A-^ i s a c o m p l e t e b l o c k s y s t e m o f G . T h e n G c H ^ . (A^jN-A-^) . S i n c e (A-^N-A^) i s a m i n i m a l ( k ) - g r o u p we h a v e G - H^A-^N-A-^) , a n d h e n c e ( a B ) - e G . . T h i s c o n t r a d i c t s t h e a s s u m p t i o n t h a t (a(3) . G . C a s e 3. G A i s o f t y p e 1 . L e t 1(A- L,A 2) c H w h e r e A 1 = l1±>±2> • • • ^ k - l ^ 3 ^ l 3 h 3 ' ' ' > J ' k - l ^ a n d A-j fl A 2 = <J) • We c o n s i d e r t h e c a s e k = 3 a n d . t h e c a s e k > 4 s e p a r a t e l y . F o r k = 3 > we f i r s t p r o v e t h a t ( i - j i p ) > ( j j J g ) e G I f n o t , ( i 1 i 2 ) ( a p ) , (j'1J2)(aP) e G a n d G c o n t a i n s e i t h e r B =' {1 , (±1±2)(j-L32) > ^ l^l) ) 3 ( i i J ' 2 ^ i 2 J ' l ^ o r C - {1 , ( i 1 i 2 ) ( j 1 j 2 ) , ( i 1 j 1 i 2 j 2 ) , ( i - ^ i g j ^ ) ) . , T h e s e a r e b o t h t r a n s i t i v e g r o u p s o n t h e s e t A-^  U A 2 . I f we c a n s h o w t h a t G n a s a 3 - c v c l e f o r e a c h c a s e , t h e n , b y t h e t r a n s i t i v i t y o f B a n d C , t h i s w o u l d i m p l y t h a t G c o n t a i n s a s u b g r o u p o f t y p e , a n d h e n c e G D . T h e r e f o r e G A i s o f t y p e S ^ w h i c h c o n t r a d i c t s t h e a s s u m p t i o n GA'~"i~s"~6Y t y p e ^ . C o n s i d e r a p a r t i t i o n P = {X-^X^X^} o n N d e f i n e d a s f o l l o w s : -X± = { i ^ } , X 2 = ti19a,tl} , X ? = N { i 2 , J 2 3 .. 17. L e t g b e a P - e l e m e n t i n G . N o t e t h a t g 4 ( a B ) . I f g i s a n y t r a n s p o s i t i o n b e s i d e s ( a B ) , t h e n b y t h e t r a n s i t i v i t y o f B a n d C o n A^ U~A 2 a n d t h e t r a n s i t i v i t y o f G o n N we c a n e a s i l y s e e t h a t G = S y m ( N ) , a c o n t r a d i c t i o n . O t h e r w i s e , we h a v e t h e f o l l o w i n g c a s e s : ( i ) g i s a 3 - c y c l e i n w h i c h c a s e we a r e d o n e . ( i i ) g = ( a p ) ( i 2 J 2 ) w h i c h i m p l i e s g ( a p ) ( i 1 ± n ) i s a 3 - c y c l e . ( i i i ) g = ( J ± a ) ( i 2 j 2 ) o r g = ( J 1 P ) ( i 2 J 2 ) . H e r e g ( i ± i 2 ) ( J X J 2 ) i s a 5 - c y c x e ^ a n d h e n c e , b y t h e t r a n s i t i v i t y o f G , G i s p r i m i t i v e . I f A l t ( N ) cj: G , t h e n b y L e m m a 2 .9 G i s o f t y p e < (126) (35.4) , (12345) , (25) (34) > . B u t < ( 1 2 6 ) ( 3 5 ^ ) . , (123^5) , (25) (3^) > i s n o t a ( 3 ) - g r o u p , a c o n t r a d i c t i o n . H e n c e G 3 A l t ( N ) . T h u s ( i ^ i g ) , ( j ^ ) e G a n d B , C d: G . T h i s i n t u r n i m p l i e s t h a t g = ( j ± a ) o r g = ( j ^ P ) b y t h e s a m e a r g u m e n t a s a b o v e . H e n c e G c o n t a i n s a s u b g r o u p o f t y p e . T h e r e f o r e , b y t h e s a m e a r g u m e n t a s <• i n c a s e 2 , G i s o f t y p e H ^ ., N o w , s u p p o s e k > i . T h e n A l t ( A ± ) a n d A l t ( A 2 ) a r e i n G . L e t h b e a P - ^ - e l e m e n t i n G w h e r e P l = t X - ^ j X g , . . . , X ^ } i s a p a r t i t i o n o n N d e f i n e d a s f o l l o w s : X ± = 3 X 2 = t O i * 0 ^ ? ) i X ^ • = { i 2 , J 2 3 > • • • j •= l - j ^ i * J ] £ _ x } I f h(A,) D A 2 4 ^ * t h e n G c o n t a i n s A l t(A-,UA p) a n d 1 8 . s o c o n t a i n s A n . . O t h e r w i s e h = ( j -^a) , ( j-^P) , (j-]_aP) o r ( j ^ P a ) . T h i s i m p l i e s G 3 S y m ( A 2 l l { a } ) , S y m ( A 2 U { P } ) o r A l t (AgUCctjP}) • S i n c e G i s . t r a n s i t i v e , t h e l a s t p o s s i b i l i t y i m p l i e s t h a t G 3 A n a n d h e n c e G A i s n o t o f t y p e ^ w h i c h i s a c o n t r a d i c t i o n , a n d t h e f i r s t t w o p o s s i b i l i t i e s e a s i l y i m p l y t h a t G c o n t a i n s a s u b -g r o u p o f t y p e H k . T h i s c o m p l e t e s t h e p r o o f . C o r o l l o r y 2.11. L e t G b e a m i n i m a l ( k ) - g r o u p o f d e g r e e n = 2k . T h e n f o r k _> 3 G i s o f t y p e H f c . P r o o f : By Theorem 2 , 1 G i s t r a n s i t i v e , T h e r e f o r e G c o n t a i n s a s u b g r o u p o f t y p e H , o r A b y T h e o r e m 2.10 K n B u t A ^ i s n o t m i n i m a l f o r k => 3 • H e n c e G i s o f t y p e H-„ • sr. C o r o l l o r y 2.12. L e t G b e a ( k ) - g r o u p i n S n w h e r e n •- 2k . T h e n G c o n t a i n s a s u b g r o u p o f t y p e S k + - ^ > A k + 2 o r H k ' §4. ( k ) - g r o u p s o f d e g r e e n = 2k + 1 . We h a v e s h o w n t h a t e v e r y t r a n s i t i v e ( k ) - g r o u p o f d e g r e e 2k + 1 i s p r i m i t i v e . F o r t h e s t u d y o f s u c h g r o u p s , we l i s t . s o m e p r o p e r t i e s o f p r i m i t i v e g r o u p s . We q u o t e t h e f o l l o w i n g t h e o r e m s -(2.13-2.17) f r o m [ 5 ] . 19-T h e o r e m 2.1j. L e t G b e a p r i m i t i v e g r o u p o n N a n d l e t (j) 4 +" N . T h e n , f o r a n y t w o d i s t i n c t a a n d p i n N , t h e r e e x i s t s g e G w i t h a e gCN^) a n d B | g ( N 1 ) ( S e e p.15 o f [5]) . T h e o r e m 2 . 1 4 . A p r i m i t i v e g r o u p o n K w h i c h c o n t a i n s a t r a n s p o s i t i o n i s t h e s y m m e t r i c g r o u p o n N . A p r i m i t i v e g r o u p w h i c h c o n t a i n s a 3 - c y c l e i s e i t h e r t h e a l t e r n a t i n g g r o u p o r t h e s y m m e t r i c g r o u p o n N ( S e e p . 3 4 o f [ 5 ] ) . T h e o r e m 2.15. A p r i m i t i v e g r o u p o f d e g r e e n w h i c h c o n t a i n s a c y c l e o f l e n g t h m w i t h 1 < m < n i ' s ( n - m + l ) - f o l d t r a n s i t i v e . ( P . J4 o f [ 5 ] ) . T h e o r e m 2.16. L e t G b e a t r a n s i t i v e p e r m u a t i o n g r o u p o f p r i m e d e g r e e . T h e n G i s s o l v a b l e i f a n d o n l y i f f o r a * P > G { a , 6 } = 1 ' ( P - 2 9 o f [ 5 ] ) ' \ T h e o r e m "2.17. E v e r y n o n s o l v a b l e t r a n s i t i v e g r o u p o f p r i m e d e g r e e i s d o u b l y t r a n s i t i v e . (P .29 o f [5]) . ' T h e l a s t t w o t h e o r e m s e a s i l y i m p l y C o r o l l o r y 2 .18. ^ T r a n s i t i v e ( k ) - g r o u p s o f p r i m e d e g r e e , w h e r e k > -j , a r e n o t s o l v a b l e , a n d h e n c e a r e d o u b l y t r a n s i t i v e . 20. L e t G = < (12345) , (13)(45) > a d i h e d r a l g r o u p o f o r d e r 10. ( I n t h e s e q u e l we w i l l c a l l s u c h a g r o u p a g r o u p o f t y p e D<_) . L e m m a 2.19- T h e o n l y m i n i m a l ( 2 ) - g r o u p s o f d e g r e e 5 a r e o f t y p e L\_ . P r o o f : L e t G = < h = (12345) , (13)(^5) > b e o f t y p e D(~ . We f i r s t s h o w t h a t ' G i s a ( 2 ) - g r o u p . L e t P = {X^,X^} b e a p a r t i t i o n o n {1,2,3,4,5) . W i t h o u t l o s s o f g e n e r a l i t y , we m a y s u p p o s e | X - J = 1 o r jX-^j = 2 . I f = [a] f o r some a 3 then by the t r a n s i t i v i t y o f G t h e r e e x i s t s g e G s u c h t h a t g (2) = a . T h e n g _ 1 ( 1 3 ) (45)g i s a P - e l e m e n t i n G . I f (x^ l = 2X , t h e n e i t h e r X± = h s ( { l , 3 } ) o r X-j^ = h s ( { 4 , 5 } ) f o r s o m e s s i n t e g e r s . T h e n h (13)(45 )h i s a P - e l e m e n t i n G . H e n c e G i s a ( 2 ) - g r o u p . N o w , l e t H \ b e a n y m i n i m a l ( 2 ) - g r o u p i n S,_ w i t h n o f i x e d p o i n t . T h e n , b y T h e o r e m 2 . 1 , H i s t r a n s i t i v e , a n d s o 5 d i v i d e s | G | . B y S y l o w ' s t h e o r e m , t h e r e e x i s t s g e H o f o r d e r 5 . We m a y w r i t e g = (12345) C o n s i d e r t h e p a r t i t i o n P = { X - ^ X g } o n {1,2,3,4,5} d e f i n e d b y X-j^ = {1,2,3} , X 2 = {4,5} . S i n c e H i s t r a n s i t i v e we c o n c l u d e , b y T h e o r e m 2.3, t h a t H i s p r i m i t i v e . I f H c o n t a i n s a t r a n s p o s i t i o n o r a 3 - c y c l e t h e n b y T h e o r e m 2.14 i t . . f o l l o w s , t h a t H i s n o t m i n i m a l . 21. T h e r e f o r e t h e o n l y p o s s i b l e P - e l e m e n t s a r e h ^ = (12)(45) , h 2 = (13) (45) a n d *i = (23) (45) . S i n c e (134) = h ±g a n d (245) = h ^ g s n e i t h e r c a n b e i n H . T h e r e f o r e h g i s t h e o n l y p o s s i b l e P - e l e m e n t i n H a n d < g , h g > i s a d i h e d r a l g r o u p i n S,- . A p e r m u t a t i o n g r o u p G o n N i s c a l l e d s e m i -r e g u l a r i f , f o r e a c h a e N , G = {1] ; a n d G i s c a l l e d r e g u l a r i f i t i s ' s e m i r e g u l a r a n d t r a n s i t i v e . L e t G b e a n o n - r e g u l a r p r i m i t i v e g r o u p o f d e g r e e 7 s u c h t h a t G d> A . B u r n s i d e h a s s h o w n i n [1] ( p . 217) t h a t G m u s t b e c o n -j u g a t e i n t o o n e o f t h e f o l l o w i n g g r o u p s <'(1234567) , (243756) > , <(1234567) , (235)(476) > , < (1234567) , (27)(45)(36) > o r < (1236457) , (234)(567) , (2763)(45) > . L e m m a 2.20. G = < (1236457) , (234)(567) , (2763)(45) > i s a m i n i m a l ( 3 ) - g r o u p o f d e g r e e 7 . P r o o f : A s i t i s s h o w n i n [ 1 ] , G i s d o u b l y t r a n s i t i v e . S i n c e (26)(73) e G i m p l i e s (73)(45) = [ (1234)(567) ]~ 1 (26)'(73) [(234) (567)] e G , t h e r e s t r i c t i o n g r o u p G ^ 1 , 2 , 5 , 6 J 7 ^ c o n t a i n s a s u b g r o u p o f t y p e H g . T h e r e f o r e e v e r y G p o n = N - {a ,6} c o n t a i n s a s u b g r o u p o f t y p e H g f o r a l l d i s t i n c t a , B e N . We w i l l s e e i n T h e o r e m 3.6 o f t h e n e x t c h a p t e r t h a t G ^ - j = < (234) (567) , (2763) (45) > i s a ( 2 ) - g r o u p v ^ a n d s o Gj a.j' l s a (2)-group f o r a l l a e N ( T h i s g r o u p i s a s p e c i a l c a s e o f d e f i n e d i n C h a p t e r I I I ) . S i 22. N o w , l e t P = { X 1 , X 2 , X ^ 3 b e a p a r t i t i o n o n N . T h e n o n e o f X ^ , s a y X ^ , h a s a t m o s t t w o e l e m e n t s . I f |X-J = 0 o r | X-^ | = 1 , t h e n a P - e l e m e n t c a n b e c h o s e n f r o m G r •> f o r s o m e a e N , b e c a u s e G r •> i s {a } > {a } a ( 2 ) - g r o u p . I f |x n | = 2 a n d X . = { a , p ) t h e n t h e r e A ft e x i s t s 1 4 h e G ' a p s u c h t h a t h ( X 2 ) _ = X g a n d h{X^) = X ^ T h i s i m p l i e s e i t h e r h o r ( a p ) h i s a P - e l e m e n t i n G . H e n c e G i s a ( 3 ) - g r o u p . T h i s c o m p l e t e s t h e p r o o f . We n o t e t h a t <(1236457) , (234)(567) , (2763)(45) i s i s o m o r p h i c t o P S L ( 2 , 7 ) . T h e r e f o r e , i n t h e s e q u e l , we w i l l c a l l s u c h a g r o u p o n e o f t y p e P S L ( 2 , 7 ) I f G i s a p r i m i t i v e g r o u p o f d e g r e e 7 a n d n o t o f t y p e P S L ( 2 , 7 ) c r , t h e n i t i s c l e a r t h a t ^^1} ^ s a s e m i r e S u l a r g r o u p a n d t h e r e f o r e i t i s n o t a ( 2 ) - g r o u p . H e n c e ^ s u c h a G i s n o t a ( 3 ) - g r o u p . I n c o n c l u s i o n , we h a v e L e m m a 2.21. T h e o n l y m i n i m a l ( 3 ) - g r o u p o f d e g r e e 7 i s o f t y p e PSL(2V,7) ' . L e m m a 2.22. L e t h , g / e S n b e s u c h t h a t t h e i r c y c l e d e c o m p o s i t i o n h a v e o n l y o n e p o i n t i n c o m m o n . T h e n h " " 1 " g " 1 h g i s a 3 - c y c l e . P r o o f : L e t a b e t h e c o m m o n p o i n t i n h a n d g , a n d l e t h ( a ) = p a n d , g ( a ) = y . T h e n h ~ 1 g " 1 h g = (Pay) . 23.. Theorem 2.23. L e t G be a '(k) -g roup o f degree 2k + 1 where k _> 4 . Then G c o n t a i n s t h e a l t e r n a t i n g g roup on N = { 1 , 2 , . . . ,2k+ l } . P r o o f : We use i n d u c t i o n on k . By Theorem 2.3 G i s p r i m i t i v e . T h e r e f o r e , b y Theorem 2 . 14 , i t i s s u f f i c i e n t t o show t h a t G c o n t a i n s a 3 - c y c l e . We s t a r t w i t h t h e case ' k = 4 . Suppose t h a t G 4 . Then G c o n t a i n s no t r a n s p o s i t i o n on N , and t h e r e f o r e G A , where A = N - {a, f3} , i s a (3 ) -g roup o f degree < 7 f o r a l l d i s t i n c t a (3 e N . Hence G A c o n t a i n s a subgroup o f t y p e ( i ) S^ , ( i i ) ( i i i ) H^ o r ( i v ) PSL(2.7). . For cases ( i ) , ( i i ) and ( i i i ) , we c a n see e a s i l y t h a t G c o n t a i n s a 3 - c y c l e . Now c o n s i d e r case ( i v ) . Assume G A = < (1236457): , .(234) (567) , (2763) (45) > . Then < (1236457)", (234)(567) ," (26)(73),>.c G . By Theorem 2.15 G i s 3 - f o l d t r a n s i t i v e . T h e r e f o r e t h e r e e x i s t s g 1 , g 2 e G such t h a t {1,4,5} £ S ] _ ( t 2 , 6 , 7 , 3 ) ) and { l , a , p } c g 2 ({2,6,7,3}) . I f g ± ({2,6,7,3} ) n {2,6,7,3} + ty f o r some i , t h e n b y Lemma 2.22 g7 1 (26) (73)gj_ (26) (73) i s a 3 - c y c l e i n G . O t h e r w i s e , g 1( {2,6,7,3}) fl g 2 ({2,6,7,33) has e x a c t l y t h r e e e l e m e n t s , and t h e r e f o r e t h e p r o d u c t g ^ " ( 2 6 ) ( 7 3 ) g 1 * g 21 ( 2 6 ) ( 7 3 ) g 2 i s e i t h e r a 3 - c y c l e o r a 5 - c y c l e . The second, p o s s i b i l i t y i m p l i e s G i s a 5 - f o l d t r a n s i t i v e . Hence we. can f i n d a n o t h e r g ^ e G such t h a t (26)(73) and g" 1 (26) (73 )g^ have o n l y one p o i n t i n common, 2 4 . -a n d . a g a i n b y L e m m a 2 . 2 2 G h a s a 3 - c y c l e . F o r k > 4 , a s s u m e t h e t h e o r e m i s t r u e f o r a l l ( k - 1 ) - g r o u p s o f d e g r e e 2 ( k - l ) + 1 . L e t G b e a ( k ) - g r o u p o f d e g r e e 2 k + 1 3 a n d l e t G =j= s k + 1 . L e t a a n d B b e t w o d i s t i n c t p o i n t s i n N a n d l e , t A •- N - { a , B 3 . T h e n G A i s a ( k - l ) - r g r o u p o n A . I f G A h a s n o f i x e d p o i n t i n A , t h e n b y t h e i n d u c t i o n h y p o t h e s i s A l t ( A ) C G . I f G A h a s s o m e f i x e d p o i n t i n A , t h e n G A i s o f d e g r e e <_ 2 k . H e n c e b y C o r o l l o r y 2 . 1 2 G c o n t a i n s a s u b g r o u p o f t y p e A k + 1 , S k + 2 o r ^ k * I n e a c h c a s e , we . n o t e ' t h a t G h a s a 3 - c y c l e . T h e r e f o r e G 3 A ^ _ , _ ^ . T h i s c o m p l e t e s t h e p r o o f . We s u m m a r i z e t h e r e s u l t s o f t h i s c h a p t e r i n t h e f o l l o w i n g t h e o r e m . T h e o r e m 2 . 2 4 . L e t G b e a ( k ) - g r o u p o f d e g r e e _< 2 k + 1 T h e n ^ ( i ) G c o n t a i n s a s u b g r o u p o f t y p e S , , Ak , E0 o r 3 ' l Dr i f k = 2 . / • 5 ( i i ) G c o n t a i n s a s u b g r o u p o f t y p e S,. , A , R% o r P S L ( 2 , 7 ) i f . k = 3 • ( i i i ) G c o n t a i n s a s u b g r o u p o f t y p e S k + 1 , \_+2 o r ^ k i f k k - 4 " : 25. . C h a p t e r I I I  C o n s t r u c t i o n a n d E x a m p l e s o f M i n i m a l ( k ) - g r o u p s . T h e a i m o f t h i s c h a p t e r i s t o c o n s t r u c t m a n y d i f f e r e n t ' t y p e s o f ( k ) - g r o u p s i n c l u d i n g t h o s e n e e d e d I n t h e f o l l o w i n g c h a p t e r . L e t G-^ a n d G 2 b e t w o p e r m u t a t i o n g r o u p s o n N-^ a n d N g r e s p e c t i v e l y . T h e W r e a l t h p r o d u c t o f G-, b y G 2 , d e n o t e d b y G j \ G 2 i s , b y d e f i n i t i o n , t h e g r o u p o f a l l p e r m u t a t i o n s g o n N-^ x N 2 o f t h e f o l l o w i n g k i n d : g ( a , P ) = (gp(a) , h ( p ) ) a e N x > p e N g w h e r e f o r e a c h P e N g , g A i s a p e r m u t a t i o n o f G^ o n , b u t f o r _ d i f f e x e n t P ' s t h e c h o i c e s o f t h e p e r m u t a t i o n g R a r e i n d e p e n d e n t . P I f G-^ a n d G 2 a r e t r a n s i t i v e , t h e n G-^ G 2 i s t r a n s i t i v e o n x N 2 . G ]_ G 2 i s n o t P r i m i t i v e i f G 2 i s n o t t r i v i a l . I n p a r t i c u l a r , i f we t a k e G ^ = S^. a n d G g = S g , a n d c o n s i d e r S^^V S g a s a p e r m u t a t i o n g r o u p o n { 1 , 2 , . . 2 k ] , t h e n S ^ ' V S 2 i s o f t y p e d e f i n e d i n C h a p t e r I I . H e n c e " \ ^ S g i s a m i n i m a l ( k ) - g r o u p . We m a y w r i t e I; H k = S k ^ - S 2 * . 26. T h e a b o v e e x a m p l e s u g g e s t s a u s e f u l a p p r o a c h t o c o n s t r u c t i m p r i m i t i v e m i n i m a l ( k ) - g r o u p s b y u s i n g t h e W r e a l t h p r o d u c t . § I . C o n s t r u c t i o n o f i m p r i m i t i v e m i n i m a l j ( k ) - g r o u p s . T h e o r e m 3 . 1 . L e t S y m ( N x ) b e o f d e g r e e i a n d S y m ( N 2 ) b e o f d e g r e e (^) + 1 w h e r e 2 <_ i <_ k . L e t G = S y m ( N 1 ) \ S y m ( N 2 ) . T h e n G i s a m i n i m a l ( k ) - g r o u p o n x. N 2 o f d e g r e e n = [ ( ^ ) + l ] - i . P r o o f : L e t jNg = { l , 2 , . . . , m ] w h e r e m = ( i ) + 1 . F o r e a c h B e N £ •, l e t = N± x £ 6 ) . T h e n { J ^ J 2 , . . , Jm3 i s a p a r t i t i o n o n N £ x N g . I t i s n o t d i f f i c u l t t o s e e t h a t G i s g e n e r a t e d b y S y m ( J p ) ' , B e N 2 a n d t h o s e p e r m u t a t i o n s w h i c h i n t e r c h a n g e t h e e l e m e n t s o f t w o o f ' t h e J Q ' , a n d k e e p t h e o t h e r p o i n t s f i x e d . N o w , l e t P = { X - ^ , X 2 , . . . , X ^ ) b e a n y p a r t i t i o n o f l e n g t h k o n N , x N 0 . • I f X . D Ja c o n t a i n s m o r e i d t t p t h a n o n e p o i n t f o r s o m e t a n d B , t h e n a P - e l e m e n t g c a n b e t a k e n f r o m S y m ( J p ) . O t h e r w i s e , we s u p p o s e l x t n J B ' - 1 f o r a 1 1 * a n d 6 * S i n c e m > ( i ) > t h e r e a r e B-^ a n d B 2 s u c h t h a t | X T n J p | = | X T fl J p | f o r a l l t . T h i s i n d u c e s a o n e - t o - o n e c o r r e s p o n d e n c e h b e t w e e n t h e N e l e m e n t s i n J0 a n d t h o s e i n J 0 3 1 27-C l e a r l y , h i s a P - e l e m e n t a n d i s i n G . H e n c e G i s a ( k ) - g r o u p . T o s h o w t h a t G i s m i n i m a l , l e t K b e t h e s e t c o n s i s t i n g o f a l l t r a n s p o s i t i o n s i n G a n d t h o s e p e r m u t a t i o n s o f o r d e r 2 w h i c h i n t e r c h a n g e t h e e l e m e n t o f t w o o f t h e J R k e e p i n g t h e o t h e r p o i n t s f i x e d . T h e n K i s a s e t . o f g e n e r a t o r s o f G . We s h a l l s h o w t h a t , • f o r e a c h g e K , t h e r e i s a p a r t i t i o n P o n N-^ x N g s u c h t h a t g i s t h e o n l y P - e l e m e n t i n G . L e t p 1 € N 2 . T h e n N 2 - {p^} h a s e x a c t l y (^) e l e m e n t s , a n d s i n c e | J R | = i f o r a l l p e N 2 , t h e r e e x i s t s a p a r t i t i o n P , = {X, , X Q , . . . , X , } o n U J f t 1 1 2 k p4px P s u c h t h a t n o P - e l e m e n t i s i n S y m ( N 1 ) \ Sym(N 2~{P^}) . Now s u p p o s e t h a t g i s a t r a n s p o s i t i o n i n S y r a ( J A ) . • P l L e t P 2 = { Y ^ , Y 2 , . . , Y i ^} b e a p a r t i t i o n o n J f t s u c h t h a t g i s t h e o n l y P p - e l e m e n t i n S y m ( J f t ) . T h e n g * p l i s t h e o n l y P - e l e m e n t i n G w h e r e P = { X ^ Y - ^ X g U Y g , . . . . . , X i _ 1 Q Y i 1 , . . . X k ) i s a p a r t i t i o n o n x N g . N e x t s u p p o s e t h a t g i s a p e r m u t a t i o n o f o r d e r 2 w h i c h i n t e r c h a n g e s t h e e l e m e n t s o f J Q w i t h t h o s e i n Ja p l p 2 L e t g = ( a , b , ) ••• ( a n . b . ) w h e r e a . e J0 a n d l i' J p-|_ b . e J f t f o r a l l j = l , 2 , . . . , i . W i t h o u t l o s s o f j ' P 2 g e n e r a l i t y we m a y a s s u m e J f t (1 X . •= {b .3 f o r j = 1 , 2 , . . . , : P g • 3 - 3 H e n c e g i s t h e o n l y P - e l e m e n t i n G w h e r e P = [X.j[}[a.^}3 . . . , X i u { a i 3 , X i + 1 , . . . , X k } i s a p a r t i t i o n o n N-^ x N 2 . S i n c e g , p^ a n d p ? a r e a r b i t r a r y , e v e r y ( k ) - g r o u p c o n t a i n e d i n the p r o o f . G must c o n t a i n t he s e t K 28. T h i s c o m p l e t e s Remark 3 . 2 . ( i ) I n p a r t i c u l a r , i f we take i = k , t h e n n = 2 k and G i s o f the t ype H"k . ( i i ) L e t rru = ( ( k ) + l ) i . Then m-± < m^ < rrig.x < • • • < m k where [ g ] i s t he \ k l a r g e s t i n t e g e r l e s s t h a n o r e q u a l t o ^ . I n f a c t , m. = ( ( k ) + l ) - i - k i + ± ~ ( k - i ) I ( i - l ) ! + ' < ( k - l ) ! , ( l - l ) . ' + k " 1 + 1 and '((k-l+l)f(i-l)l + 1 } (k-1+1) = ^ - i + l f o r 1 < [ | l M i + 1 " = ( ( 1 ^ 1 ) + l ) ( i + l ) - ( ( kki + 1 ) + l ) ( k - i + l ) • = ( ( k - i - 1 ) ) i i + ( ( k - i ) l ( i - l ) ! + k - 1 + 1 ) ( k - i ) - k l i - k ! -g- y - u ( k - i ) ! ( k - i ) l i J N = (k~2i) •..iUk-Dl + 2 1 " k -. > k - 21 - ( k - 2 i ) = 0 f o r i - < - J k ] • 29. Thus f o r d i s t i n c t v a l u e s o f i , .the correspond-i n g (k)-groups are not isomorphic. C o r o l l o r y 3 . 3 : Let G be an i m p r i m i t i v e t r a n s i t i v e group of degree n < ( ( 2 ) + l ) - 2 w i t h complete b l o c k system {C,,C Q,...,C } where |C. | < k . Then G i s not a (k)-group. Proof: Let jC.| = t and m = ( ( ^ ) + l ) - t . Then G ————— i jj can be regarded as a subgroup of S. "\_ S . Since Tr m k k n < ( ( g ) + l ) . 2 <_ ( ( t ) + l ) - t , G i s p r o p e r l y c o n t a i n e d i n S..\ S . But S, \ S i s a minimal (k)-group. T h e r e f o r e t ^ m t m v G i s not a (k)-group. Theorem 3 . 4 . Let Sym(N 1) be of degree k and l e t H be a s e m i - r e g u l a r subgroup on N 2 of degree s such t h a t H i s o f prime o r d e r . Let G = Sym(N 1)\H . Then G i s a minimal (k)-group o f degree sk . Proof: Let N 2 = { 1 , 2 , . / . ,s} and J± = N x { i ) , i = l , 2 , . . . , s . L e t P = { X 1 , X 2 , • • >\) b e a p a r t i t i o n on ^ x N 2 • I f i X^nJ± | >_ 2 f o r some i and j , then a P-element g can be chosen from Sym(J i) . Otherwise, | X . H J . | = 1 f o r a l l i = 1 ,2 , ...,s , j = 1 , 2 , . . . , k . T h e r e f o r e a--. P-element e x i s t s i n G by the p r o p e r t i e s o f Wrealth product. Hence G i s a (k)-group. 30. To show G i s m i n i m a l , l e t K be any ( k ) -g roup c o n t a i n e d i n G . L e t (aB) be a t r a n s p o s i t i o n i n Sym(J^.) . We c o n s t r u c t a p a r t i t i o n P = {X-^,Xg, . . . *X^} on x N 2 as f o l l o w s : X 1 f l J t = [a>»&) > Xg n J t = <|> and |X R J . | = 1 f o r a l l ( j , i ) £ {1 .2 } x { t ) . I t i s c l e a r t h a t (aB) i s t h e o n l y P-e lement c o n t a i n e d i n G and hence (aB) e K . T h e r e f o r e S y m ( J 1 ) x . . . x S y m ( J s ) c K . Bu t s i n c e Sym(J- ] ) x . . . x Sym(J ) i s n o t a ( k ) - g r o u p and s i n c e H i s s e m i r e g u l a r o f p r i m e o r d e r , we have G = K . T h i s c o m p l e t e s t he p r o o f . L e t { N - j ^ N g , ] ^ } be a p a r t i t i o n on N such t h a t J N j = k f o r a l l i . L e t B = o {g 6 Sym(N). 1 g i s e v e n , g(N ±) = N. l f o r a l l 1 } B l - {g e Sym(N) g i s o d d , g(N 2) = g(N 3) = Ng] Bg = {g € Sym(N) g i s o d d , g ( N ± ) = N 3 • g ( N 3 ) = B 3 = {g £ Sym(N) g i s o d d , g ( N 2 ) = N2,, g(N 2) = •\) {g Sym(N) g i s e v e n , g ( N . ) = = N . 0 , g ( N j ) -• N t g ( N ^ ) = N ± , where ( i , j , t ) = {1 ,2,3} I t i s n o t d i f f i c u l t t o check t h a t t he u n i o n G = B U . . .U B o i s a g roup and B Q i s a n o r m a l subgroup o f G . A l s o , f o r each g . e B. , B g . = B. , i = 1 , 2 , 3 , and hence ° i i 3 o 1 l 3 G = < B o ' S i ' 6 2 ' s 3 > • W e P r o v e ; \ -Theorem 3 . 5 . The g r o u p G d e f i n e d as above i s a m i n i m a l ( k ) - g r o u p on N o f degree 3k . 5 1 • Proof: Let P = {X^Xg, . . . ,X^} be a p a r t i t i o n on N . We have the f o l l o w i n g three p o s s i b l e cases: Case 1 . I f there e x i s t s i ^ , i g , and. J 2 , j- 4 j- , such t h a t |X. ON, | > 2 , |x. ON. | > 2. , then ( a 1 a 2 ) ( 3 1 P 2 ) , f o r some a 1 , a 2 e X i n Nj , P 1 , P 2 e X ± fl i s a P-element. Case 2. I f there e x i s t s e x a c t l y one N. , say N-. , such t h a t |X.nNn I > 2 f o r some t , then j X.HN . | = 1 f o r a l l i = l , 2,...,k and j = 2,3 . Thus there e x i s t s a permutation g on N such t h a t (a) g(a) = a f o r each a e N x , (b) g(p) = y , g(y) = P whenever p e N 2 , Y e NL and p , y e X. f o r some X. . Thus g le a v e s 2 1 1 each X. i n v a r i a n t . We note t h a t g i s odd i f k i s odd, and t h e r e f o r e g i s a P-element i n G . On"the o t h e r hand, i f k i s even, then (ap)g e B-^  i s a P-element i n G f o r . some a , p e X t fl N-^  . Case 3- I f |X.nN.| = 1 f o r a l l i and j , then c l e a r l y we can choose a P-element from B^ c G . Thus G i s a (k)-group. To show G i s minimal, l e t H be a (k)-group c o n t a i n e d i n G . For d i s t i n c t J x > J 2 e {1,2,3} and x± , x g e Nj , y 1 , y 2 e N.^ , l e t N o = txi>x2> - , , x k 5 ' N j = Cy i a y 2 *- - - -»y k } a n d N - (N0- U N ^ ) = {z-^, z 2 , . . ., z^} . L e t P = {X-^ , ...jX^.} be a p a r t i t i o n on N d e f i n e d by X-^  = { x 2 . * x 2 , z i ^ * 2 = { y i * y 2 , z 2 ^ * x 3 = [^-jf'yj)'zj)j . x k = ( xk>yk> zk} • 3 2 -Then i t i s c l e a r t h a t ( x ^ x ^ ) ( y x y 2 . ) i s t he o n l y P-e lement i n G and hence ( x x x 2 ) ( y x y 2 ) e H . S ince J ' l ' J*2  3 x l ' x 2 a n d y l » y 2 a r e a r b l t r a r y i w e have B Q c H . On the o t h e r h a n d , f o r t h e p a r t i t i o n P = { Y ^ , . . . on N d e f i n e d b y Y± = {x 1,x 2,y 1,z 1} , Y 2 = Cy2,z2] , Y^ - { x ^ y ^ z ^ Y k = we f i n d t h a t H n B . 4 ^ • S ince j n i s a r b i t r a r y , we J l x have H fl B. | f , i = 1 , 2 , 3 . T h e r e f o r e H = G and hence G i s m i n i m a l . I n a manner s i m i l a r t o t h e p r o o f o f Theorem 3 - 5 , we may p r o v e Theorem L e t B and B,. be d e f i n e d as above £ — o 4 For each i = 1 , 2 , 3 , l e t B i = { g e Sym(N) | g i s even g ( N g ) = N t , g ( N t ) = N s , where { s , t } = {1 ,2,3} - {!}'} . Then G = B Q U ! ] _ U 5 2 U U B'^  i s a m i n i m a l . ( k ) - g r o u p o f degree 3k . I n the s e q u e l we w i l l c a l l t he g r o u p G i n Theorem 3 .5 and G i n Theorem 3-6 a g roup o f t y p e and r e s p e c t i v e l y . I n t h e n e x t c h a p t e r , we w i l l show t h a t g roups o f t y p e S^X ' %: a n d ^ k a r e o n - 1 - y p o s s i b l e m i n i m a l ( k ) - g r o u p s o f degree 3k . 3 3 -§ 2 Construction of i n t r a n s i t i v e minimal (k)-groups with no fixed point. Let G be any group and m a po s i t i v e integer. Let ^ m(G) denote the set of a l l normal subgroups K such that G/K i s of order m . Suppose that (J)2 : G-j/K-^  = G 2/K 2 , K l e ^ m ^ l ) 3 K 2 € Jm ( Q 2 > ' L e t f i : G i "* G i / K i ' i = l j 2 be the natural homomorphisms. We define G = {gjS^ I £>xe&±>8>2e(^2 3 ( j ) 2 ( f 1 ( g 1 ) ) = f* 2(g 2)} • I t i s not d i f f i c u l t to see that G i s a group under the operation -(g-^) (h-jhg) = (g 1h 1) (gghg) . We c a l l G the star product of G^ and G 2 with respect, to (K^,(j)^) , i = 1 , 2 where <J)^  i s the i d e n t i t y map on G-j/K^ . We denote i t by" (G 1,K 1,d) 1) * (G2,K2,4>2) . Note-that the set {g-j_g2 Igje^jggeKg} = x K 2 = K i s a normal subgroup of G and K e ? m ( G ) • More generally, f o r each s 2 2 , we can define an s-star product of , i = I , 2,...,s to be G= { g - j g g . • S s l s i € G i ^ i ( f l ( g l ) ^ = f i ^ S i ^ 3 1 = 1J2*--->s1 and denote i t by .f-, \ (G. ,K. ,<J). ) where K. e 3 m ( G i ) a n d ((h : G 1/K 1 = G j / K i f o r a 1 1 1 = 2 , 3 , - . . s , and ^ i s the i d e n t i t y map on . Again, K = {g^gg.-gs ^ € 1 ^ , 1 = 1 , 2 , .. ,s} = K, x . .x K„ € 7 (G) . One may see e a s i l y that the star 1 s mv ' product * i s associative and commutative. Theorem 3 - 7• Let G^ be a minimal (k)-group on , i = l , 2 , . . . s , and l e t K. e 3" (G. ) f o r a l l i , where P 3 3 N 1 P 1 i s a prime. Let ty^ - i d e n t i t y may on G^/K^ a n d (J^ : G-^ /K^  = Gi/K1. , !•;= l , 2 , . . . , s . Then the s-star 34 . p r o d u c t G = i f 1 * ( G i , K i , ( j ) ^ ) i s a m i n i m a l ( k ) - g r o u p on t h e s e t N = A N. . 1=1 1 P r o o f : S ince t h e * p r o d u c t i s a s s o c i a t i v e , b y u s i n g i n d u c t i o n on S , we need o n l y t o p r o v e t h e case s = 2 . L e t G^/K^ = < g K x > and G 2 / K 2 = <hK g > where gK^ and h K 2 a re c o s e t s . L e t P = { X - j ^ X g , . . . , X k ) . be a p a r t i t i o n on N ± U N g . Then P± = [ N ^ n X ^ , . . . ^ N ^ n x ^ } and P 2 = { N g f l X ^ , . . . , N 2 n X k } a re p a r t i t i o n s on and N 2 r e s p e c t i v e l y . By t h e assumpt ions o f G ^ and G 2 t h e r e e x i s t P^ and P 2 - e l e m e n t s g^ and g 2 r e s p e c t i v e l y . I f e i t h e r g^ e o r h^ € K 2 t h e n we a re done. O t h e r w i s e , g 1 = g g 2 and h^ = h h 2 , where g 2 e K ± , h 2 e K g and : m. t.. (m,p) = ( t , p ) = 1\ . T h e r e f o r e g - ^ = g g ^ and = h h ^ f o r some i n t e g e r s m^ and t ^ , and g ^ e , h^ c K 2 . T h i s i m p l i e s ( g g ^ ) ( h h ^ ) i s a P-e lement i n G . Hence G i s a ( k ) - g r o u p . To show t h a t G i s m i n i m a l , l e t H be any ( k ) - g r o u p c o n t a i n e d i n G . S ince G ^ i s m i n i m a l , l e t P l = t x i j X 2 * * ' * > X k^ b e a P a ? ' f c l ' t l o n o n N i such t h a t no ? 1 - e l e m e n t i s i n K-^  . Fo r each P 2 = { Y ^ , Y g , . . . , Y ^ } on N 2 , P = [ X ^ U Y , . . . , X k U Y k ) i s a p a r t i t i o n on N-^ U N 2 and hence t h e r e e x i s t s a P-e lement g i n G . L e t p 2 T be the subgroup G = g e n e r a t e d b y t h e s e t { g | P p P 2 c-i s any p a r t i t i o n on N p - o f l e n g t h k } . S i n c e G p i s N m i n i m a l , i t i s easy t o see t h a t t h e r e s t r i c t i o n map T G g 35-i s onto and hence K 2 — T — G ' S i m i l a r l y , c T . However, s i n c e ^ x K g i s not a k-group and s i n c e G/K^xKg i s of prime o r d e r , we have H = G . Hence G i s minimal. Theorem 3-7 g i v e s a u s e f u l method of c o n s t r u c t i o n o f i n t r a n s i t i v e minimal (k)-groups. For example, l e t G^ and G 2 be two minimal (k)-groups on and N 2 r e s p e c t i v e l y , such t h a t the subgroup G^ of even permutations i n G± i s proper. Then G? e ? 2(G j L) f o r i = .1,2, and t h e r e f o r e there i s o n l y one isomorphism <})2 : G^/G^ = G 2/G 2 . Hence ( G ^ G ® , (j^) * (G 2, G| , <t>2) , which we w i l l denote by G^ * G 2 , i s a minimal (k)-group.. We n o t e ' t h a t ^ j r ^ * ( G ^ K ^ , ^ ) i s not a (-k)-group i f any one o f G i i s not a (k)-group. Furthermore, K, e 3" (G.) > i = 1.2,. . ,s , where P i s a prime, i s a S ' n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r ^K^* (G±»^-±> $±.) t o be a (k)-group whenever G^ i s a minimal (k)-group f o r a l l i . Indeed we have Theorem 3-8. Let G.^  be a minimal (k)-group on N i , and K± e ^ ( G ^ , i = l , 2 , . . . , s such t h a t ^ : G ^ / K ^ = f o r a l l i . I f m i s not a prime, then G = i f 1 ( G i j K ^ t j h ) i s not a (k)-group. *• X Proof: We need o n l y a proof f o r the case s = 2 k 36. Let f. : G. - G./K. , i = 1,2 be the n a t u r a l 1 l r l ' ' homomorphism and l e t L-^  be a proper subgroup o f G-^ /K^  . Then f^" L(L- L) i s a proper subgroup of G^ . Since G^ i s minimal, we can choose a p a r t i t i o n = {X^,X 2,.. ,Xk} on such t h a t no P^-element i s i n • L e t M-j^  be the s e t of a l l P^-elements i n G^ wit h I d e n t i t y . -Then i s a proper subgroup i n G^ and hence M 2 = f 2 (pgf-^CM^) i s a proper subgroup i n G^ having t r i v i a l i n t e r s e c t i o n w i t h f " 1 ^ ^ ) . Let P 2 = {Y 1,Y 2,..,Y. be a p a r t i t i o n on N 2 such t h a t no P 2~element i s i n M 2 . Then, f o r the p a r t i t i o n P = {X 1UY- L,X 2UY 2, . . . ,X kUY k} on N-^  U N 2 , there e x i s t s no P-element i n G . Hence G i s not a (k)-group. Remark 3-9- I f we omit the c o n d i t i o n t h a t a l l G. are minimal, then m = p , a prime, i s no lon g e r a n e c e s s a r y c o n d i t i o n f o r G = . f * (G. ,K. ,(j). ) to be a (k)-group. For example, take G^ = Sym(N 1) and G 2 = Sym(N 2) where N 1 = {1,2,3,4} and N g = {5,6,7} , and % = {1 , (12)(34) , (14)(23) , (13)(24)} , K g = {1} . Then <J)2 : G-^ /K^  = G 2/K 2 and G = (G-^K^,^) * (G2,K2,<|)2) i s generated by the s e t {(12)(56) , (34)(56) , (13)(67) , (24)(67)} • We s h a l l show t h a t G i s a • (2)-group. For each a , f5 e N 2 , G A with A = N]L U N 2 C V- {a,p} i s of type H 2 on-the s e t A . Thus g i v e n a p a r t i t i o n P = {X^X } on N 1 U N 2 , one of X^ n N g and X g D N g has more 37-than one element, say a , B e X x D N 2 , and hence there e x i s t s 1 4 g € G A such t h a t gfX^.nN^) = X± fl N and g(X 2DN 1) = X 2 n . T h i s i m p l i e s t h a t one o f g and (aB)g i s a P-element i n G . Hence G i s a (2)-group. Furthermore, i t i s c l e a r t h a t G i s minimal. ! C o r o l l o r y 3.10. An isomorphic image of a ( I t )-group i s not n e c e s s a r i l y a (k)-group. Proof: I t f o l l o w s from Remark 3-9. s i n c e G x i s a (3)-group and the r e s t r i c t i o n map ty : G -» G^ i s an isomorphism, but G i s not a (3)-group. Theorem J>.11. Let and N 2 be two s e t s such t h a t N ± n N 2 = ty and |Nx| = |Ng| = k 2 + 1 , and l e t ty : N± - N g be a one-one map. Let G be a permutation group generated by the s e t {(aB) (<j)(a)({)(p)) | a , B e N, ) . Then G = S P \ x k +1 2 and i t i s a minimal (k)-group of degree 2(k +1) . Proof: Let P = {X-^Xg, . .. ,X^} be a p a r t i t i o n on N = N x u N 2. . Since |Nx| = |Ng| = k 2 + 1 , one of X ± 0 N x , say X x fl N x , has more than k-elements, and t h e r e f o r e X. D N p c o n t a i n s a t l e a s t two elements from the s e t (t)(X xnN x) . f o r some j . Let ty(a) , ty(&) € Xj n N g n ty (X xnN x) Then c l e a r l y ^ (aB) {ty(a)ty(B)) i s a P-element i n G . Hence G i s a (k)-group. 38. To show G i s minimal, l e t H c G and suppose (a0) (<j)(a)(|)(0)) t H f o r some a , ft e H . Choose a p a r t i t i o n P.^  = {X-^Xg, . . . ,X k) on such t h a t a , p e X 1 IX-J = k + 1 and \X^\ = k f o r i 4 1 > a n d a p a r t i t i o n P 2 = {Y 1,Y 2, ...,Y k) on N 2 with ty(a) , <J>(0) e Y x and |(J)(X.)nY.| = 1 f o r a l l (i,, j ) 4 • W e combine P and P 2 to get a p a r t i t i o n P = {X ^ Y - ^ .. . jX^UY^.} on N = N 1 U N 2 . I t i s easy to see, t h a t (ap") ((j)(a)<j)(0)) i s the o n l y P-element i n G and hence H i s not a ( k ) -group. Hence G i s minimal. Theorem 3.12. Let N-^  and N 2 be two se t s such t h a t N- n N 2 = $ a n d 1^1 = |Ng| = k 2 + 2 , and l e t <J) : N x -' Ng be a 1-1 map. L e t ^ .G be generated by the s e t £ (apv) (<()(a)(|)(p)<J)(Y)) la^YeN-^} . Then G i s isomorphic 2 to A p and i t i s a minimal (k)-group o f degree 2(k +2) k +2 Proof: Let P =\{X^,Xg,...,X^} be a p a r t i t i o n on N = N 1 U Ng . Then, e i t h e r ( i ) X^ . n N x has more than k + 1 elements f o r some j , /or ( i i ) |X. ON, | = |X. DN, | = k J- d2 -1 J-^ J-f o r some j g + • C a s e ( i ) . I f f o r some i ^ , X.^  n Ng c o n t a i n s a t l e a s t three elements from <J)(X. PIN,)., say (j)(x, ) , §(x0) and (J)(x^) , then ( x ^ g X ^ ) ((^(x^j^)()).(Xg)())(x^)).... i s a P-element i n Otherwise, there are i„- and i , such t h a t X. n N 0 \ 3 . "^2 C. D N 9 both c o n t a i n two elements from <t)(x^ HN, ) . 1 3 ' J l • 1 and X39. L e t (|)(y1) , ty(y2) e X ± D N g fl (J) ( X j n ^ ) and (|)(z1) , (|)(z2) e X ± n N 2 D ^ ( X j n N x ) . Then ( y 1 y 2 ) ( z i z 2 ) ( < l ) ( y i ) ? ( y 2 ^ ^ ( z 1 ) < l , ( z 2 ) ) i s a P-element i n G . Case ( i i ) Let i g be such t h a t | (X . nN 2 , )n^)(X. 0 ^ ) 1 >2 s = 2 ,3 . Let (j)(x1) , (j)(x2) e ( X ± nN 2) n ty(X. m±) and <$>(¥]_), ty(Y2) e <j)(Xj n ^ ) n (X . nN 2) . Then ( x 1 x 2 ) ( y 1 y 2 ) ( ^ ) ( x 1 ) ( | ) ( x 2 ) ) ( 0 ( y 1 ) < | ) ( y 2 ) ) i s a P-element i n G . Hence G i s a (k)-group. The m i n i m a l i t y o f G can be shown as i n the pro o f o f Theorem 3 .11 . Theorem 3.11 and Theorem 3.12 p r o v i d e two simple examples o f i n t r a n s i t i v e minimal- (k)-groups. I n the f o l l o w i n g , we s h a l l g e n e r a l i z e these r e s u l t s to t c o p i e s . To do t h i s , we need Lemma 3.13. Let N be a s e t o f k^ elements. Then there are t p a r t i t i o n s P. = {X. , X . . 1 » i = l»...jt on N such t h a t . r u X . /. N has e x a c t l y one element f o r any mapping a : { l , 2 , . . , t } - {l , 2,...,k} . Proof: L et N = vi where I = { 1 , 2 , . . . , k } . t Let X i j = ( xi> • •'• > xt) I x j [ = J a n d x s i s a r b i t r a r y i n I f o r s =j= i } f o r ^ i = 1 ,2,. . . , t and j = , 1 ,2,. . . ,k . ho. For each i = l , 2 , . . . , t , l e t P. be the p a r t i t i o n t 1 {X. X. , } . Then , n n X . /.\ c o n s i s t s o f one L i , l i . k J 1=1 i , a ( i ) p o i n t , namely, ( a ( l ) , . . . a ( t ) ) f o r any mapping a : [1,2,...,t} - (1,2,..,k} . Theorem J.lh. Let , N 2 , . . . , N t be set s o f k* + 1 elements and l e t ^ : -• , i = l , 2 , . . . , t be one-to-one maps. Let G be the group generated by the s e t {(c0)(4>2(a)<|>2O)) . . . (<|>t(a)<|>t(B)) | a , 0 e N±) . Then G i s isomorphic to S . and i t i s a minimal t k + 1 t (k)-group on N = . y,N. o f degree t ( k +1) . Proof: L et -P = {X^Xg, . . . ,X^} be a p a r t i t i o n on N . Since |N.| = k* + l ' f o r i = l , 2 , . . . , t , one of X^ fl N , 1 a x say X. D N-, has a t l e a s t k^~ X + 1 elements. T h e r e f o r e . 1 (X. ON,) 0 ^ ( X . 0N P) has a t l e a s t k t - 2 + 1 elements J x i i J 2 f o r some j 2 . By r e p e a t i n g the above step, we see t h a t -i \ -i (X. PIN,•). n <j)~ (X. PIN0) fl ... fl <j)7X(X. p|Nc) has a t l e a s t Jj^ -1- <=• J2 "s k t _ s + 1 elements. In p a r t i c u l a r , (X. PIN, ) fl _1 '' J l .. D (j). (X. DN, ) has a t l e a s t two elements. L e t ^ t a , p e X. n N x n ... fl ^ ( X j nN t) , Then (aB) (4>2(a)4>2(0)) .... ((j>t (a)<t)t(p)) i s a P-element i n G . Hence G i s a (k)-group. To prove G i s minimal, i t s u f f i c e s to show N t h a t f o r each a , P e N ± , there e x i s t s a p a r t i t i o n P on N such t h a t (ap) (<J)2 (a)<f>2 (p)) . . . ($ t (a)(J)t (p)) i s 41. the only P-element i n G . By Lemma 3 . 1 3 , there are pa r t i t i o n s P. = {X. ,,...,X. ,} , i = l , 2 , . . . , t , on X X ^ X 1 Nn - {al such that . ruX. ,. > has exactly one element J. K ' 1=1 l , a [1-) f o r a l l a : . {1 ,2,...,t} - {1,2, ... ,k} . Let X U {a} i f B e X X 3 J . X 3 J t f o r i = l , 2 , . . . , t and j = l , 2,...,k . Then . PUY. ,.\ has exactly one element i f a £ Y. /.\ f o r some i , £ i »o ) and . ruY. \ has exactly two elements i f a e Y. 1=1 x,o{i) " i , a ( , i ; f o r a l l i . I f we take (J^  : -• to be the i d e n t i t y . map, then P.7 = {(b. (Y. .., ) , ... , (j). (Y. . )} i s a p a r t i t i o n 1 1 1 , 1 1 1 , XL ^. on N. f o r a l l i = 1 , 2 , . . . , t . Hence P = {. IJ-, (Y. . )} , X ^. X — X o X ^ AC ... , ^ U ^ t ^ i k ^ i s a P a r t i t i o n on N . I t i s not d i f f i c u l t to see that (aB) ( ( { ^ ( a ) ^ ^ ) ) . . . (§ t (a)(j>t (3)), i s the only P-element i n G . Hence G i s minimal. By using a s i m i l a r proof, we may show, Theorem 3 . I 5 . Let , N2.-, ... , N g be sets of k + 2 elements and l e t (j)^  : , i = 2 , 3 , . . . , t be one-one mappings. Let G be the group generated by the set £(aPY)(<|)2(a)<|)2(P)(|)2(Y)) (4>t (a)<j>t (S )<j>t (y)) | o,S,Y e N} Then G i s isomorphic to A , and i t i s an• i n t r a n s i t i v e k + 2 t minimal (k)-group of degree t(k +2) . Chapter IV 4 2 . (k)-groups o f degree n , 2k + 2 ^ n 3k The purpose o f t h i s c h apter i s to f i n d a l l the (k)-groups o f degree n where 2k + 2 <_ n <_ 3k . §1. Ins t r a n s i t i v e minimal (k)-groups o f degree n , 2k + 2 ^  n v< 3k" . Lemma 4 . 1 . L e t G be a minimal (k)-group on N with o r b i t s N, , :Np , ... , N p . Then, f o r each i , X . N. e i t h e r N i i s a s i n g l e t o n o r G i s a ( k ) - g r o u p s . Proof: We may assume no N. i s a s i n g l e t o n . Then N. N. G 1 i s not t r i v i a l f o r a l l i . Suppose G 1 i s not a N l .(k)-group f o r some i ^ , say G . We show that G^ i s a (k)-group. Let P, = {X, ,X P, . . . ,X, } be a p a r t i t i o n on N, such t h a t no P,-element i s i n G , and l e t s P 2 = {Y 1,Y 2,...,Y k) be any p a r t i t i o n on ^Ug^i * T n e n P = [X 1UY 1,X 2UY 2,...,X kUY k) i s a p a r t i t i o n on N , and hence there e x i s t s a P-element i n g 6 G . E v i d e n t l y , g e G„ . Hence G w i s a (k)-group c o n t a i n e d i n G , w l • 1 N1 N. a c o n t r a d i c t i o n . Hence G i s a (k)-group f o r a l l i 43< Lemma 4 . 2 . Let N, c N , |N, | > 5 . Let G be a N l p e rmutation group on N such t h a t G = Alt(N- L) o r Sym(N 1) and G N_ N 4 i1) • A l t c G . Proof: L et H = {g e G | g(N,) = N,} . Then the 1 1 N-N, k e r n e l o f the n a t u r a l homomorphism H -• H = H/H^. ^  i s not t r i v i a l by the assumption. The k e r n e l H N N c o n s i s t s a l l elements o f G f i x i n g every l e t t e r i n N - N , N N On the ot h e r hand G = H = H/H„ a n d by N, al the n a t u r a l homorphism H -• H = H/H^ the normal sub-group Hjy i s mapped i n t o a normal subgroup ^ /H of H s H§ne§ . N-N x % 1 / ' H N 1 = HN-N 1/ HN-N 1 0 H N 1 \ i s a n o n - t r i v i a l normal subgroup o f G = A l t ( N 1 ) o r Sym(N 1) , and hence the subgroup H N_ N o f G i s A l t ( N 1 ) or Sym(N 1) . Theorem 4 . 3 - Let G be a minimal i n t r a n s i t i v e ( k ) -group o f degree n , 2k + 2 < n < 3k . Then n = 2k + 2 4 4 and G i s of type S k + ]_ * S k + 1 . Proof: By Lemma 4 . 1 , G has only two orbits N, , N N Ng , and G and G are k-groups. Case ( i ) . n = 2 k + 2 . This implies |N,| = |N0| = k + N N and G 1 = Sym(N-L) , G 2 = Sym(N2) . Let N l = ^ x l ' " ' , , x k + l ^ a n d N 2 = ^ 1 ' • *-' yk+l^ ' C o n s i d e r t h e p a r t i t i o n P = {X1,...,Xk) defined as follows: X-_ = Cx 1,x 2,y 5) , X 2 = Cy 1,y 2,x 5) , X^- {x4,y4] X k = t x k + l , y k + l 5 ' Then i t i s easy to see that the P-element i n G must be ( x 1 x 2 ) ( y 1 y 2 ) . Since x± , x g , y± , y^ are ar b i t r a r y , we have A l t ( N ± ) , Alt(Ng) c G . Hence G i s of type S k + I * S k + 1 " Case ( i i ) . n 4 2 k + 2 r. This implies n > 9 and so one of |N.| , i = 1 , 2 i s greater than 4 , say N N ? |Nn | > k + 2 > 5 . Since G and G are t r a n s i t i v e z N N (k)-groups, we have A l t ( N 1 ) c G and Alt(Ng) c We note that A l t ( N ± ) fl G 4 { 1 } This i s obvious i f IN.J > |N2| . On the other hand, i f IN.J = |N0| , then, 2 1 since n _< Jk < 2 (k + 1 ) f o r a l l k , by Theorem 3 - 1 1 , A l t ( N 1 ) fl G 4 tl} • Hence A l t ( N 1 ) c G by Lemma 4 . 2 , i . e . G contains a (k)-group of type 2 . This contradicts the minimality of G . Hence G does not exi s t . '" ' . ' 45-C o r o l l o r y 4.4. Let 2k + 2 < n £ 3k and l e t G be a minimal (k)-group o f degree n . Then G i s t r a n s i t i v e on N . § 2 . T r a n s i t i v e (k)-groups o f degree n where 2k + 2 x< n 3k - 2. Theorem 4 . 5 . Let 2k + 2 < _ n _ < 3 k - l , k _> ^  and l e t G be a t r a n s i t i v e (k)-group of degree n . (1) I f G i s not p r i m i t i v e , then G has a complete N ' N bl o c k system {N^,Ng) such t h a t G and G are ( k ) -groupg. ( i i ) I f n i s odd, then G i s p r i m i t i v e . .r Proof: We note t h a t ((g)+l)2 = k 2 - k + 2 _ > 4 k - 2 > 3 k f o r k 2 4 . Suppose G i s not p r i m i t v e and l e t [N-^Ng, .. . ,N } be a complete b l o c k system o f G . I f s > 3 , then IN-J <_ k f o r a l l i = l , 2 , . . . , s . Therefore by C o r o l l o r y 3 . 3 G i s not a (k)-group, which i s i m p o s s i b l e . N Hence s = 2 . Now, i f G / i s not a (k)-group, then N 2 by the t r a n s i t i v i t y o f G , n e i t h e r i s G a (k)-group. Choose p a r t i t i o n s P ] [ = {X-^Xg,. . . ,Xk} and Pg = {Y-^Yg,...^ on . N, and N D r e s p e c t i v e l y , such t h a t no P.-element N. 1 i s i n G 1 , i = 1,2 . Since I N J = |Ng| < 2k , there e x i s t s a t l e a s t one p a i r , - say X and Y. wit h |X, | 4 l Y n I \ and t h e r e f o r e f o r the p a r t i t i o n P = {X 1UY 1,XgUYg,...,X kUY k) on N , there i s no P Telement i n G . T h i s c o n t r a d i c t s 46. N the assumption t h a t G i s a (k)-group. Hence G a n d G are (k)-groups and t h i s proves ( i ) . ( i i ) f o l l o w s immediately from ( i ) . C o r o l l o r y 4 . 5 . Let n be even and 2k + 2 <_ n <_ 3k -k >_ 4 . Let G be an i m p r i m i t i v e t r a n s i t i v e (k)-group o f degree n . Then A l t (N-^) A l t ( N g ) c G where {N-^N •} i s a complete b l o c k system o f G . Furthermore, Sym(N 1)*Sym(N 2) c G i f n = 2k + 2 . Proof: By Theorem 4 . 5 G has o n l y a complete b l o c k N N 2 system {N-.,Np} , and G and G are (k)-groups. N N Therefore A l t ( N 1 ) c G 1 and Al t ( N g ) c G 2 . Let H = {g € G | g(N x) = N^} . We show t h a t H i s a ( k ) -group. Suppose the c o n t r a r y . Then there i s a p a r t i t i o n P = {X^Xg, . . . ,X^} on N such t h a t no P-eiement i s i n H . Let Y i = X. n ^  and Zj = X j D Ng , i , j = 1 , 2 , . . Since n <_ 3k - 1 » there e x i s t s i ^ and j ^ such t h a t |Yi | 4 lZj I • Thus f o r some arrangement o f i g and j g there i s no P-element i n G/ f o r the p a r t i t i o n P, = {Y. UZ. ,...,Y. UZ. } , which c o n t r a d i c t s the 1 xl J l x k J k assumption t h a t G i s a (k)-group. Hence H i s a •- 2 (k)-group. Now, s i n c e n < 2(k +1) , G N i s not t r i v i a l f o r i = 1,2 by Theorem 3 . 1 1 . Hence A l t ( N ± ) c G f o r i = 1,2 by Lemma 4 . 2 . T h i s completes the p r o o f o f the f i r s t p a r t o f the c o r o l l o r y . Furthermore, f o r n = 2k + 2 H c o n t a i n s Sym(N 1)*Sym(Ng) .by Theorem 4 . 3 . 47-Theorem 4.7. Let G be a p r i m i t i v e t r a n s i t i v e (k)-group o f degree n where n = 2k + t and 2 <_ t _< k - 2 Then G c o n t a i n s A l t ( N ) . Proof: We use i n d u c t i o n on t to prove t h i s theorem. F i r s t we c o n s i d e r the case t = 2 . Since G i s p r i m i t i v e , i t s u f f i c e s to show t h a t G has a 3 - c y c l e . I f not, A l t ( N ) £ G and hence f o r any three d i s t i n c t a , ( 3 , Y i n N , Sym({a,R,Y}) PI G = {1} . Let A =N - {a,p,Y} • Then G A i s a (k - 1 )-group of degree <_ 2 (k-1) + 1 , and hence by Theorem 2 . 2 4 G c o n t a i n s a subgroup of type ( i ) PSL ( 2 . 7 ) p r o v i d e d t h a t k = 4 , ( i i ) A ^ J . - L 3 ( i i i ) o r ( i v ) ^ . We propose to show th a t G has a 3 - c y c l e f o r each case which i n f a c t would c o n t r a d i c t ' t h e above assumption and g i v e the d e s i r e d r e s u l t . Case ( i ) . L et <(1236457) , (234)(567) , (2763)(45) > c G . Then (26)(73) and (1236457) are i n G . The l a t t e r i m p l i e s (by Theorem 2 . I 5 ) t h a t G i s 4 - f o l d t r a n s i t i v e and hence there i s g e G f o r which ( 2 6 ) ( 7 3 ) g ~ 1 ( 2 6 ) ( 7 3)g i s a 3 - c y c l e . Case ( i i ) . Let A l t ( A ' ) c G A' where | A ' | = k + 1 . Let H = {g±g2 e G I g x 6 Sym([a,p,Y}) and g 0 e A l t ( A ' ) } . C l e a r l y the k e r n e l o f the r e s t r i c t i o n map : H - Sym({a,0,Y}) i s not t r i v i a l . T herefore H f , | ( l j and hence by Lemma 4 . 2 , A l t (A') c H £ G . Case ( i i i ) . F o r k _> 4 , t h i s reduces to case ( i i ) - . 48. S o we may suppose k = 4 , i . e . n = 10 . Let A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } and A 1 = { 1 , 2 , 3 , 4 } be such t h a t S y m ( A 1 ) c G A . I f A l t ( A ) ^  G , then G c o n t a i n s < (1.2) (exp) , (34),(,ap) , (13) (PY) , (24)(pY)> ( r e f e r to Remark 3-9)- Consider the p a r t i t i o n P = {Y-^Yg,Y^,Y^} on N where Y± = {1,2,a} , Y 0 = {3,P,Y} , Y ? = {4} , = {5,6,7} . Le t g be a P-element i n - G . We c l a i m t h a t g ( A ) = A . I f , g ( A 1 ) f l (N-A) + ty , then G ( N " A ) u A l i s a t r a n s i t i v e group o f degree 7 and hence G has a 7 - c y c l e , and t h i s reduces to case ( i ) . Hence g ( A ) = A , and t h e r e f o r e g = (12)h , g = (12)(py)h o r g = (Py)h where h e Sym({5,6,7}) . I f h i s o f o r d e r 3 , then the square o f g i s a 3 - c y c l e i n G . I f h i s o f order 2 or i d e n t i t y , then, (12)h e G i m p l i e s t h a t (12)h(13)(pY) - (123)(pY)h € G and so (123) e G , (12 ) (p Y)h e G i m p l i e s t h a t ( 1 2 ) (py)h ( 1 3 ) (PY) = (123)h e G and so (123) e G , and f i n a l l y (PY)h € G i m p l i e s t h a t (p Y)h(12) (aP) = (aPy)x(12)h e G and so (aPy) e G . Thus, i n each case, G co n t a i n s a 3 - c y c l e . Case ( i v ) . L e t A-^ = {x^,x 0, . . . and A 2 = ^1^2' ' ' ' , y k 1^  b e t w o s u t , s e , t s o f N such t h a t H k l ^ l 3 ^ — G ^ " S i n c e G n a s no 2 - c y c l e or 3 - c y c l e , G c o n t a i n s ( x 1 x 2 ) h 1 , ( x 0 x ^ ) h 2 and ( y 1 y 2 ) h ^ where each h^ i s a 2 - c y c l e from Sym({a,p,y}) • Note t h a t h-^  f h 2 and t h e r e f o r e h , i s d i f f e r e n t from h^ or h 0 , say h.^  . Then the square o f the product ( x 1 x 2 ) h 1 ( y 1 y 2 ) h ^ i s a 3 - c y c l e . 4 9 -N o w , l e t G b e a p r i m i t i v e ( k ) - g r o u p o f d e g r e e 2 k + t w h e r e 3 <_ t <_ k - 2 , a n d . a s s u m e t h a t t h e t h e o r e m i s t r u e f o r a l l p r i m i t i v e ( k ) - g r o u p s o f d e g r e e 2 k + t ^ w h e r e 2 <_ t ^ <_ k - 2 a n d t 1 < t . I f A l t ( N ) £ G , t h e n S y m ( {a,P,y}) n G = { 1 } f o r a l l d i s t i n c t a , p , y e N . L e t A = N - { a , p , y } • T h e n G A i s a ( k - 1 ) - g r o u p o f d e g r e e <_ 2 ( k - l ) + t - 1 . L e t H b e a m i n i m a l ( k - 1 ) - g r o u p c o n t a i n e d i n G . I f t h e d e g r e e o f H i s l e s s t h a n 2 ( k - l ) + 2 f,(it t h e n , b y T h e o r e m 2 . 2 4 , H i s o f t y p e ( a ) S k , ( b ) A k + 1 o r ( c ) H k _ 1 . I f t h e d e g r e e o f H i s g r e a t e r o r e q u a l t o 2 ( k - 1 ) + 2 , t h e n T h e o r e m 4 . 5 a n d C o r o l l o r y 4 . 6 s h o w s t h a t H i s e i t h e r i n t r a n s i t i v e o r p r i m i t i v e . B u t , b y t h e i n d u c t i o n h y p o t h e s i s a n d t h e m i n i m a l i t y . o f H , we k n o w t h a t t h e l a t t e r i s i m p o s s i b l e . T h e r e f o r e H i s i n t r a n s i t i v e , a n d h e n c e b y T h e o r e m 4 . 3 H i s o f t y p e ( d ) $ k * S k • F ° r t h e c a s e s ( a ) , ( b ) a n d ( d ) , G h a s a 3 - c y c l e b y L e m m a 4 . 2 . F o r t h e c a s e ( c ) , we m a y u s e t h e s a m e a r g u m e n t a s t h a t i n c a s e ( i v ) a b o v e , t o s h o w t h a t G h a s a 3 - c y c l e . T h e s e a r e a l l i m p o s s i b l e . H e n c e A l t ( N ) c G . § 3 I m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p s o f d e g r e e n = 3 k . L e m m a 4 . 8 . L e t G b e a t r a n s i t i v e ( 2 ) - g r o u p o f d e g r e e 6 w i t h c o m p l e t e b l o c k s y s t e m { N - ^ N g } . T h e n G c o n t a i n s a s u b g r o u p o f t y p e S - ^ ^ S ^ a n d h e n c e G i s n o t m i n i m a l . 50. Proof: Let JH - [1 ,2 ,3} , = {4,5 ,6} and Sym(N i) £ G , i = 1,2 . We may assume t h a t (12) £ G and c o n s i d e r the p a r t i t i o n s P ± = {X X,X 2} and P 2 = {Y ±,Y 2} on N d e f i n e d by X x = {1 ,2,4} , X 2 = {3 ,5 ,6} Y1 = {1,2,6} , Y 2 = {3 ,4,5} • L e t g 1 , g 2 e G be P-^  and P 2-elements r e s p e c t i v e l y . p Then ( g x g 2 ) ~ i s - a " 3 - c y c l e on N 2 . Hence A l t ( N i ) c G , i = 1,2 . On the othe r hand, Sym(N i) cj: G i m p l i e s g 1 = (12)(56) . Hence G c o n t a i n s a subgroup o f type V s ? • Lemma 4 . 9 - Let G be an i m p r i m i t i v e t r a n s i t i v e minimal (k)-group o f degree n = 3k . Then G has o n l y one complete b l o c k system {N^H^N^} . Proof: " Lemma 4 . 8 shows t h a t the lemma i s true f o r k = 2 . For k = 3 , n = 9 j, i t i s obvious. So we suppose k 2 4 and hence by C o r o l l o r y 3 .3 G has a complete b l o c k system {N^,N2,...,Ng} where s = 2 or s = 3 • T h e r e f o r e , i f k i s odd, we are done. We. show t h a t s = 3 a l s o when k i s even. Suppose not, i n which case s = 2 . Let N x = {x x,x 2,...,x n^ 2} . and N 2 = {y x,y 2,..•,y n/ 2} • Since Sym(N.) | 6 i = 1,2 , we may assume t h a t (x_x„) £ G . 51-T h i s i m p l i e s t h a t G A ,. where A = N - { x - ^ X g ^ ^ } , i s an i n t r a n s i t i v e (k - 1 )-group of degree <_ 3(k - 1 ) , and each o r b i t has l e n g t h l e s s than 2(k - 1 ) . Hence G A c o n t a i n s a subgroup o f type S k , A k + 1 o r S k * S k ' i ' e * G A c o n t a i n s a subgroup o f type A^ . . We know t h a t f o r each g € G A , e i t h e r ( x 1 x 0 ) g o r g i s i n G . Therefore G c o n t a i n s a subgroup o f type and hence c o n t a i n s A l t ( N 1 ) and A l t ( N g ) . T h i s c o n t r a d i c t s the m i n i m a l i t y of G . Hence s = 3 . Theorem 4 . 1 0 . Let G be an i m p r i m i t i v e t r a n s i t i v e minimal (k)-group o f degree n = 3k . Then G i s of type S k \ A 5 , \ ' o r ^ . Proof: ' By Lemma 4 .9 we have a complete b l o c k system {N-^Ng,^} . L e t N x = {x-^Xg, . . ,^3 , Ng = {y-^Yg, • v , y k ) N5 = l i l ' t e ' t k ) • S u P P ° s e Sym(N i) c G. f o r i = 1 ,2 , 3 . Choose a p a r t i t i o n = {X-^Xg,...,X^} on N wit h X l = ^ l ' ^ l ^ l ^ 3 X 2 = ^•x2>^2>z2^ 3 ' ' ' 9 X k = ^ ^ k ^ k ^ k ^ ' Let g^ be a -element i n G . Then g£ = 1 , f o r i f 2 g 1 = 1 , then < g 1 , Sym(N i) , i = 1 , 2 , 3 > c o n t a i n s a proper subgroup o f type H k , and s i n c e . H k i s a (k)-group t h i s c o n t r a d i c t s the m i n i m a l i t y o f G . There f o r e < g 1 , Sym(Nj), i = 1 , 2 , 3 > i s a group o f type S k"VA^ , and s i n c e S k X A ^ i s a (k)-group, G i s o f type S k \ A . 52. Now, suppose Sym(N i) G f o r i = 1 , 2 , 3 . We may suppose (x-^Xg) G . For any two elements y , y ' i n N 2 , we may form a p a r t i t i o n P 2 = [Y-^Yg, . . . ,Yk} on N as f o l l o w s : Y 1 = { x 1 , x 2 , z 1 ) , Y g = [y,y',Zg} and each Y^ f o r i _> 3 has one element from each o f the N^ . Let g 2 be a P^-element i n G . Then e i t h e r g 2 = (yy') or g 2 = ( x ^ x 2 ) ( y y ' ) . By forming products o f the d i f f e r e n t g 2 obtained, from d i f f e r e n t [y,y'} i t i s c l e a r t h a t we can o b t a i n any 3 - c y c l e (YiYjYs) a n d t h e r e f o r e Alt(Ng)- £ G Then (yy') G f o r any p a i r y., y' and so gg = ( x 1 x 2 ) ( y y / ) By t r a n s i t i v i t y of G we have A l t (N-^) c G . Hence Sym(N 1)*Sym(N 2) c G . S i m i l a r l y , Sym(N-^)*Sym(N^) c G and Sym(N2)*Sym(N^)'' c G . Next, l e t P-, = [Z, ,...,Z, } be a p a r t i t i o n on 3 J- k ^ N g i v e n by z1 = [ x 1 , x 2 , y 1 , z 1 ) , Zg = [ y 2 , Z g } and Z i = ix±>V±>z±} f o r 1 = 3,4,...,k . Let g^ be a p -element i n G . Then g^ = ( y x z x ) ' ( yk zk^ o r s 3 = ^ x l x 2 ^ y l z l ^ ' ' ,x ^ y k z k ^ " T h e r e f o r e G c o n t a i n s B 1 . i f g^ i s odd and G c o n t a i n s B^  i f g^ i s even, where B^ and B-^  are d e f i n e d as i n Theorem 3 .5 and Theorem 3-6 r e s p e c t i v e l y . S i m i l a r l y , G c o n t a i n s B 2 or Bg , and G c o n t a i n s B , o r B", . Note'that i f G c o n t a i n s B, 3 3 1 •and B"j . f o r ' 1 + J , then G 3 ' B i - B j - B i r> ^ s a n d G = Bj-Bg-B^ 3 BjL where- s € [ 1 , 2 , 3 } - { i , j } . Hence G p r o p e r l y c o n t a i n s a subgroup of type , t h i s c o n t r a d i c t s the m i n i m a l i t y of G . Therefore G c o n t a i n s a l l the B^1s or a l l the B j / s . T h i s completes the p r o o f . 5 3 -§4 P r i m i t i v e ( k ) - g r o u p s o f degree n = 3 k - 1 o r 3 k . Lemma 4 .11. L e t G be an s - f o l d t r a n s i t i v e g roup o f degree n > 2 s - 1 . I f t h e r e e x i s t s 1 | g e G such t h a t t he number o f p o i n t s i n N f i x e d b y g i s g r e a t e r t h a n o r e q u a l t o s - 1 , t h e n A l t ( N ) c G . P r o o f : L e t A be t h e s e t o f p o i n t s f i x e d by g and l e t A ' = N - A . Then | A ' | <_ s . By t h e s - f o l d t r a n s i t i v i t y o f G , t h e r e e x i s t s h e G such t h a t h ( A 7 ) fl A ' has e x a c t l y one e l e m e n t . Hence b y Lemma 2 . 2 2 t he commuta tor o f g and h _ 1 g h i s a 3 - c y c l e i n G . M o r e o v e r , s i n c e t h e e x i s t e n c e o f g i m p l i e s i t h a t s > 2 , G \ i s p r i m i t i v e . Hence A l t ( N ) c Gv . Lemma 4 .12. L e t G be a p r i m i t i v e ( 3 ) - g r o u p on N o f degree 9 . Then A l t ( N ) c G . P r o o f : We n o t e f i r s t t h a t i f G c o n t a i n s a t r a n s p o s i t i o n , a 3 - c y c l e , o r a 5 - c y c l e , t h e n G c o n t a i n s t h e a l t e r n a t i n g g roup on N . I n t h e f i r s t two cases t h e c o n c l u s i o n f o l l o w s f r o m theorem 2.14. I n t he l a s t c a s e , G i s 5 - f o l d t r a n s i t i v e (see Theorem 2 . 1 5 ) and hence t h e c o n c l u s i o n f o l l o w s f r o m Lemma 4 .11. Suppose A l t ( N ) £ G , so t h a t G c o n t a i n s no t r a n s p o s i t i o n o r 3 - c y c l e . L e t A = { a , B , Y } c N and A ' l e t A ' = N - A . Then G i s a ( 2 ) - g r o u p o f degree 54. at most 6 and hence G A contains a subgroup of the type ( i ) D 5 , ( i i ) , ( i i i ) S ? , (iv) H 2 , (v) Sg^VJ^ , (vi) Q2 , ( v i i ) ^ 2 or ( v i i i ) s^ *Sj) • T h e remainder of the proof consists of showing that i n each of the 8 cases above, G contains either a transposition, a 3-cycle, or a 5-cycle. This w i l l by the f i r s t paragraph, contradict our assumption that Alt(N) G . Case ( j ) . In this case G contains a 5-cycle and t h e r e f o r e G contains an element of the form g = g-^ gg where g-^  e Sym'( £a,p,y 3) and g 0 i s a 5-cycle on A ' . 2 3 Then one of g , g , g i s a 5-cycle i n G . A' Case ( i i ) , Let A l t ( A 1 ) c G where we may assume = {1,2,3,4 3 • Cl e a r l y Alt(A-^) £ G , and so we may write ( 123)g 2 ,(243)g 0 e G where g 1 and g 2 are 3-cycles i n Sym({a,P,Y3) • Note that i f g.^  =j= g 2 then g l = g' 1 , and therefore (143) = ( 1 2 3)g 1(243)g 2 € G . Hence we may suppose that (123)(apy) € G and (143)(aPY) € G Let P = {X-L,X2,X^3 be a p a r t i t i o n on N given by X x = £1,2,a) , .X2 = {3,PVY} and X^ = {4,5,63 where {5,63 = I*' - ^ . Let g be/a P-element i n G . I f g({l,2 , 3 3 ) n A ty then G N r ^ 5 , 6 , i ^ i s a primitive group of degree 6 . Hence G has a 5-cycle by Lemma 2.9, and. we have case ( i ) . Therefore g({l,2,33) n A = ty . Then g = h-jhgh^ where e {1,(12)} , hg e £l,(py)} • -and e . {1,(45),(46)/56)3 . Since the squares of -( 2 3)(ap Y)h^= (123)(aPY)(12)h 5; and (123)(ay)h ; 5 = (123)(apy)(PY)h^ are 3-cycles, we have, g = (12)(py)h^ . 55-By the same argument a p p l i e d to the p a r t i t i o n P' = {Y1,Y0,Y^) d e f i n e d by Y± = {2,3,a) , YG = {1,P,Y} , Y? = {4,5,6} , we have a P'-element g' = (23)(PY)h^ where h^ e {1, ( 4 5 ) , (56) , (46)} . We observe t h a t i f h ^ i s the i d e n t i t y or of order 2 , then the square ,of gg' i s a 3 - c y c l e i n G . Otherwise, we may w r i t e gg' = (123)(456) and t h e r e f o r e G A c o n t a i n s a p r i m i t i v e group o f degree 6 . Hence by Lemma 2 .9 G c o n t a i n s a 5 - c y c l e . Case ( i i i ) . L et Sym(A 0) £ G A where A 2 = {1 ,2,3} . Since A l t ( A 2 ) G , we may suppose t h a t (12)(ap) , (23)(Py) e G Consider the p a r t i t i o n P = {X-^Xp^X^} g i v e n by X ^ = { l , 2 , a ] , X 2 = C3,P,Y) , X ^ = {^^5'^} • B y same argument as i n case ( i i ) we f i n d t h a t the P-element g i s of the form (12)(pY)g-5 where' g^ e [ 1 , ( 4 5 ) , (56) , (46)} , and t h e r e f o r e (123)g^ = (23)(PY)g i s i n G . T h i s i m p l i e s t h a t ( 1 2 3 ) € G . Case ( i v ) . Let A 1 = { 1 ,2 ,3 ,4} and l e t H g( { 1 , 2 } , { 3 , 4 } ) c G . Then we have C = {(12) (ap) ,(34) (ap) ,(12)(34) } c G . Since G i s p r i m i t i v e , by Theorem 2 . 13 , there, i s g e G with a e g(A-j_) and P ^ g(A-^) • We have the f o l l o w i n g cases: / . (a) I f g ( A 1 ) n A 1 = (j) , then [ g " 1 (12) (34)g(12) (ap) f i s a 3 - c y c l e . (b) I f g(A^) n A-^  has e x a c t l y one element, then, by Lemma 2 . 2 2 , G has a 3 - c y c l e . (c) I f g^A-,) 0 An has" e x a c t l y 2 elements, we may suppose t h a t 5 e g(A-,) , then, s i n c e G f R V R fil i s 56. • t r a n s i t i v e on [ 1 , 2 , 3 , 4 } , G ^ ^ i s a t r a n s i t i v e group on {l , 2 , 3 , 4 , 5,a , 0 } . Therefore the order o f G i s d i v i s i b l e by 7 and hence G c o n t a i n s a 7 - c y c l e . Then, by Theorem 2 .15 , G i s 3 - f o l d t r a n s i t i v e . L et h 1 and hg be such t h a t {a,B,y} c h^ (A-^) and {a,5,6} c h g ( A 1 ) I f one of h 1 ( A 1 ) fl A ± , hg ( A 1 ) n A^ has e x a c t l y one element, then we are done by Lemma 2 . 2 2 . Otherwise, h^(A- L) D hg(A^) has 3 elements and t h e r e f o r e h^ x ( 1 2 ) ( 3 4)h 1hg 1 ( 1 2 ) ( 3 4)h 2 i s e i t h e r a 3 - c y c l e o r a 5 - c y c l e . (d) I f g ( A ± ) D A ± has e x a c t l y three elements, then g " x ( 1 2 ) (34)g(12) (34) i s e i t h e r a ^ 3-cycle or a 5 - c y c l e i n G . Case (v). L et H be a group o f type Sg"VA^ contained i n G A . Let {N1,Ng,N^} be a complete b l o c k system o f H with N x = {1,2}. , N 2 = {3,4} , N^ = {5,6} . Then (12)h , (34)h , (56)h e G f o r some t r a n s p o s i t i o n h i n Sym(A) , and hence \ = { ( 1 2 ) ( 3 4 ) , ( 1 2 ) ( 5 6 ) , ( 3 4 ) ( 5 6 )} c G . Let A ± = { 1 ,2 ,3 ,4} . Then there e x i s t s g e G such t h a t 5 € g ( A x ) and 6 £ g ( A x ) . 'For |g(A x) n A-J = 0 ,1 or 3 , we o b t a i n that G has a 3-cycle o r a 5 - c y c l e by same . argument as th a t i n case ( i v ) , so we_need to c o n s i d e r o n l y the case |g(A ±) 0 A-J = 2 . Assume g ( l ) = 5 . We c o n s i d e r the two cases g(2) G A.,_ -and g(2) f ^ . I f g(2) € A^ , then we may suppose g (2) = 1 and g~ 1 ( 1 2 ) ( 3 4)g equals e i t h e r 5 7 -( 5 1 ) ( a 2 ) o r ( 5 1 ) ( a 3 ) • The f i r s t p o s s i b i l i t y i m p l i e s t h a t ( 1 2 ) ( 5 6 ) ( 5 1 ) ( a 2 ) i s a 5 - c y c l e i n G . The second p o s s i b i l i t y i m p l i e s t h a t ( 2 6 ) ( 5 1 ) = [ ( 1 2 ) ( 5 6 ) ( 5 1 ) (aj>) ]2 i s i n G , and so { ( 1 2 ) ( 3 4 ) , ( 5 6 ) ( 3 4 ) , ( 1 2 ) ( 5 6 ) , ( 1 5 ) ( 6 2 ) , ( 1 6 ) ( 2 5 ) } c G . T h i s reduces t o case ( i v ) . I f , on t he o t h e r h a n d , g ( 2 ) k A x , t h e n g ( 3 ) , g ( 4 ) e A x . I f ( g ( 3 ) g ( 4 ) ) = ( 1 2 ) or ( g ( 3 ) g ( 4 ) ) = ( 3 4 ) , t h e n ( 5 g ( 2 ) ) ( g ( 3 ) g ( 4 ) ) ( g ( 3 ) g ( 4 ) ) ( 5 6 ) = ( 5 6 g ( 2 ) ) i s a 3 - c y c l e i n G . O t h e r w i s e , we may -suppose g ( 3 ) e N ± and g ( 4 ) e Ng , and i n p a r t i c u l a r g ( 3 ) = 1 , g ( 4 ) = 3 • Then ( 1 3 ) ( 2 4 ) = [ g " 1 ( 1 2 ) ( 3 4 ) g ( 1 2 ) ( 3 4 ) ] 2 e G , and hence £ ( 1 2 ) ( 5 6 ) , ( 3 4 ) ( 5 6 ) , ( 1 2 ) ( 3 4 ) , ( 1 3 ) ( 2 4 ) , ( 1 4 ) ( 2 3 ) ) c G . T h i s reduces t o case ( i v ) . Case ( v i ) ( ( v i i ) ) : . L e t H be a g roup o f t y p e Qg/ A 1 ( r e s p e c t i v e l y C g^) c o n t a i n e d i n G . L e t £ N - L , N 2 , N ^ ) be the comp le te b lock , sys tem o f H w i t h N-j^ = £ 1 , 2 } , N g = { 3 , 4 } , N^ = [ 5 , 6 } . We need o n l y show t h a t t he s e t C x = £ ( 1 2 ) ( 3 4 ) , ( 3 4 ) ( 5 6 ) , ( 1 2 ) ( 5 6 ) ] i s i n G , w h i c h g i v e s case ( v ) . I f f a c t ( 1 3 2 4 ) h ( r e s p e c t i v e l y ( 1 3 2 4 ) ( 5 6 ) h ) i s i n G where h e { 1 , ( a B ) , ( B y ) , ( a y ) } , and hence ( 1 2 ) ( 3 4 ) i s i n G . S i m i l a r l y , i t i s easy t o see t h a t ( 3 4 ) ( 5 6 ) , ( 1 2 ) ( 5 6 ) e G . Hence C± c G . Case ( v i i i ) . L e t Sym(A j ) *Sym(A^) c G A where A^•= { 1 , 2 , 3 } and A 4 = £ 4 , 5 , 6 } . Then ( 1 2 3 ) g ± , ( 4 5 6 ) g g f ( 1 2 ) ( 4 5 ) g ^ e G where g ^ and gg a r e 3 - c y c l e s , and g ^ i s t h e . i d e n t i t y o r a - t r a n s p o s i t i o n . B u t , i f g ^ = 1 , t h e n t he square o f ( 1 2 ) ( 4 5 ) g , ( 1 2 3)g - , i s a 3 - c y c l e i n G . 58. Therefore we may suppose that (123) (a0y), (456) (apy)» (12)(45)(ap) e G . Now, c o n s i d e r the p a r t i t i o n P = {X^XgjXj} on N d e f i n e d by X x = {l , 2 , 4,y} > X2 = {2>5>6} > X^ = {a,p} . Let g be a P-element i n G : I f g ( A - j ) fl A^ or g(A^) fl A-^  i s not empty, then i t f o l l o w s that (1) < (123)(465) , g _ 1 (123)(465)g > i s a p r i m i t i v e group of degree 6 i f y i s f i x e d by g , (2) < (123)(465) , g _ 1 (123)(465)g > i s a p r i m i t i v e group o f degree J i f y i s not f i x e d by g . In the case (1) G has a 5-c'ycle ( r e f e r to [1], p. 215)• In the case (2) o^ 'P) is primitive with (12) (45) e G ^ a ^ ' and hence G^a>^>} \ i s e i t h e r o f type , or PSL(2,7) ( r e f e r to [1], p. '217). But s i n c e PSL(2,7) c o n t a i n s a subgroup o f type Q2 , t h i s reduces to case ( v i ) . Hence i t remains to c o n s i d e r the case i n which g(A^) n A^ and g(A^) fl A^ are both empty. Note t h a t i f y i s not f i x e d by g i n t h i s case, say g(y) = 1 , then {(123)(aPy) , g _ 1 (123)(a0y)g} generates a p r i m i t i v e group o f degree 6 , and hence G has a 5-cycle./ T h e r e f o r e g = h^h^h^ where h x e {1,(12)} , h 2 e {1,(56)} and h } e ,£l,(o0)} . I f h 2 = 1 , then h 2 ^ 1 and h 7 | 1 , and hence (12)(45)(ap)g i s a t r a n s p o s i t i o n i n G . I f h 2 | 1 , then c l e a r l y the square o f (12)(45)(ap)g i s a 3-cycle i n G . Th i s completes the p r o o f . We quote the f o l l o w i n g r e s u l t s from Burnside [1] (p. 179 and p. 217). -59-Theorem 4.13- L e t G be a t r a n s i t i v e g roup o f degree n . I f A l t . ( N ) c G , t h e n G i s a t most [ £ ] + 1 - f o l d t r a n s i t i v e , where [ j ] i s t h e l a r g e s t i n t e g e r l e s s t h a n o r e q u a l t o ^ . Lemma 4 . 1 4 . L e t G be a n o n - r e g u l a r p r i m i t i v e g r o u p o f degree 8 . I f A l t ( N ) £ G and |G| i s d i v i s i b l e b y 7 , t h e n G i s one o f the f o l l o w i n g t y p e s : (a) < (15642378) , (1234567) , (243756) > (b) < (I627)(5438) , (1234567) , (235)(476) > ( c ) < (18) (26) (37) (45) , (1236456) , (234) ( 5 6 7 K (2763) (45) (d ) < (81 ) ( 2 6 ) ( 3 7 ) ( 4 5 ) , (1236457) , (234)(567) > (e) < (81) (26)^37)(45) , (1236457) > . Remark 4 . 15 . S ince < (1236457) > (234)(567) , (2763)(45) > i s a ( 3 ) - g r o u p , < (81 ) ( 2 6 ) ( 3 7 ) ( 4 5 ) , (1236457) , (234)(567) , (2763)(45) > i s a ( 3 ) - g r o u p . We show t h a t t h i s i s \ t h e o n l y ( 3 ) - g r o u p among the g roups l i s t e d i n Lemma 4 . 1 4 . Fo r i f n o t , g ^ * 5 J 6 , 7 , 8 } ± q a g r o u p o f degree <_ 5 , and t h e r e f o r e c o n t a i n s a subgroup o f t y p e L\_ , H 2 , A^ o r S^ . The f i r s t case i m p l i e s G has a 5 - c y c l e and t h e r e f o r e G i s a 4 - f o l d t r a n s i t i v e . Hence G f R , i s n o t t r i v i a l f o r a l l a , 0 , y e N . I n the l a s t t h r e e cases G has some e lemen ts w h i c h f i x a t l e a s t f o u r p o i n t s i n N -. Hence G r 0 „•> 1 F, 1 f o r X l c x ,p ,Y i f l±J some a , 0 , Y € N . B u t t h i s i s i m p o s s i b l e , because 6 0 . G r o v•> = { 1 ] f o r a l l g roups G l i s t e d i n Lemma 4 . 14 [ct, p , Y i e x c e p t < (81 ) ( 2 6 ) ( 3 7 ) ( 4 5 ) > (1236457) > (234)(567) , (2763)(45) Lemma 4 . 1 6 . L e t G be a p r i m i t v e ( 3 ) - g r o u p o f degree 8 . Then e i t h e r G 3 A l t ( N ) o r G i s o f t h e t y p e < (81 ) ( 2 6 ) ( 3 7 ) ( 4 5 ) , (1236457) , (234)(567) , (2763)(45) > P r o o f : We n o t e f i r s t t h a t i f G c o n t a i n s a t r a n s p o s i t i o n , a 3 - c y c l e , o r a 5 - e y c l e , t h e n G c o n t a i n s t he a l t e r n a t i n g g roup on N . I n t h e f i r s t two c a s e s , t h e c o n c l u s i o n f o l l o w s f r o m Theorem 2 . 1 4 . I n t he l a s t c a s e , G i s 4 - f o l d t r a n s i t i v e , and t h e r e f o r e b y Theorem 4 . 1 3 , A l t ( N ) c G . Now, suppose t h a t : Sym(N) (j: G and l e t A = [ 1 , 2 , . . ,63 • Then G i s a (2.) - g r o u p o f degree <_ 6 , and hence c o n t a i n s a subgroup o f t y p e ( i ) , ( i i ) A^ , ( i i i ) , ( i v ) S 5 * S 3 . ( v ) H 2 , ( v i ) Q2 , ( i i ) Q2 o r ; ( v i i i ) S 2 \ _ A ^ . Fo r the cases ( i ) - ( i v ) , we c a n e a s i l y show t h a t G has a \3 -cyc le o r a 5 - c y c l e , and hence G 3 A l t ( N ) by t h e above a rgument . Now we c o n s i d e r t h e / c a s e ( v ) . L e t A^ = { 1 , 2 , 3 , 4 3 and H 2( { 1 , 2 3 , { 3 , 4 3) c G A . I f ' G . has a 4 - c y c l e , t h e n G 3 A l t ( N ) b y Theorem 2.15 and Theorem 4 . 1 3 . O t h e r w i s e , t h e s e t C = { ( 1 2 ) ( 3 4 ) , ( 1 4 ) ( 2 3 ) , ( 1 3 ) ( 2 4 ) , ( 1 2 ) ( 7 8 ) , ( 3 4 ) ( 7 8 ) 3 c G . By Theorem 2.13, ' t h e r e i s a g e G w i t h 7- e g(A ±) and >8 1 g(A^) / I f |g(-A1)n'A1| = 1 o r | g(A 1) fl A-J = 3 , t h e n t he square o f ( 1 2 ) ( 3 4 ) g - 1 ( 1 2 ) ( 3 4 ) g i s e i t h e r a 3 - c y c l e o r a 5 - c y c l e . ;' I f |g(A-^) fl A - j J 2 , t h e n C U C S g e n e r a t e s 61. a t r a n s i t i v e group o f degree 7 . Therefore |G| i s d i v i s i b l e by 7 , and by Lemma 4 . 14 and Remark 4 . 1 5 , t h i s i m p l i e s that G i s A l t ( N ) o r o f the type < (18) (26) (37) (45) , (1236457) , (234) (567) (2763) (45) > . The p r o o f s f o r the cases (v) - ( v i i ) are e x a c t l y the same as the cor r e s p o n d i n g cases i n Lemma 4 . 1 2 . Lemma 4.17- Let G be a minimal t r a n s i t i v e ( 3)-group ; o f degree 8 . Then G i s o f type S2"\^S^ . Proof: Lemma 4.15 shows t h a t G i s not p r i m i t i v e . Let {N-pNg, . .. ,N } be a complete b l o c k system o f G . Then s = 2 or 4 . I f s = 2 , we l e t = { 1 ,2 ,3 ,4} and assume (12) G . Consider a p a r t i t i o n P = {X^,X2,X } on N d e f i n e d by X± = { 1 , 2 ^ } , X 2 = { 3,y 2,y 3} = {4,y^} , where y ± e N 2 f o r i = 1 , 2 , 3 , 4 . Let g be a P-element i n G . Then g = ( 1 2)(y 2y^) or " g = ( y 2 y ^ ) • Since x y 2 and y^ are a r b i t r a r y i n N g , we must have Al-t-fN-g-} G G as y^ and y 2 run over a l l elements i n N 2 . By the t r a n s i t i v i t y o f G we a l s o have A l t ( N x ) G G . Now s i n c e Sym(N 2) £ G , g - ( 1 2)-(y 2y 5) and hence G p r o p e r l y c o n t a i n s a subgroup o f type S^*S^ . This i s (3)-group and hence G i s not minimal. Thus s = 4 , and hence by Theorem 3-1 G i s o f type S 2~VS^ . \ Theorem 4 . 1 8 . Let G be a p r i m i t i v e (k)-group o f degree n = 3k - 1 where k > 4 . Then A l t ( N ) c G . 62. Proof; I t i s s u f f i c i e n t to show t h a t G c o n t a i n s a 3-cycle o r a 5 - c y c l e . We use i n d u c t i o n on k to prove t h i s theorem. We f i r s t show t h a t i t i s true f o r k = 4 . Suppose Sym(N) £ G . L e t .N = [1,2,...,11} and A = {1,2,...,9) . Then G A i s a (3)-group o f degree £ 9 , a ^ d by the pr e v i o u s r e s u l t s , G c o n t a i n s a subgroup o f type ( i ) , ( i i ) S^ , ( i i i ) , ( i v ) S 4 * S 4 , (v) S 3 \ A 3 , ( v i ) , ( v i i ) <Z3 , ( v i i i ) PSL(2 , 7 ) or ( i x ) Sg^jS^ . Each o f the groups i n ( i ) - ( v i i ) c o n t a i n s a 3-cycle and hence G c o n t a i n s a 3-cycle. In case ( v i i i ) , G has a 7 - c y c l e and t h e r e f o r e i t i s 5 - f o l d t r a n s i t i v e . By Theorem 4.13. Th i s happens o n l y i f A l t ( N ) c G . oHence we need o n l y to c o n s i d e r the case ( i x ) . N L e t H be a subgroup o f type S 0 v s ^ cont a i n e d i n G A and l e t {{1,2],{3,4},{5,6},{7,8}} be a complete b l o c k system f o r H . Then K = < (12) (10 11) , (34)(10 11) , (56)(10 11) , (78)(10 11) > c G , Since n = 11 i s prime, G i s doubly t r a n s i t i v e by'Theorem .2.16 and Theorem 2 Therefore there e x i s t s a g e G such t h a t g(10) = 10 and g ( l l ) = 9 • We may suppose 11 £ g({l,2}) . Then g _ 1(12)(10 l l ) g f i x e s 11 , and so f o r some h e K , the square o f h g _ 1(12)(10 , l l ) g i s (9 11 10) . Hence A l t ( N ) C G . Now, suppose k _> 5 • L e t A = {a,0,Y} £ N and l e t A ' = N - A . Then G A i s a (k-1)-group o f A ' degree o 3 ( k - l ) - 1 . I f G i s p r i m i t i v e , then by the A ' i n d u c t i o n h y p o t h e s i s , we have A l t (A ' " ) £ G and hence 63-A ' G has a 3-cycle. I f G i s not p r i m i t i v e , then, by Theorem 2.24, Theorem 4.3 and C o r o l l o r y 4.6, G A c o n t a i n s a subgroup o f type (a) , (b) , (c) s k * s k o r ( d) \ i • Each of the groups i n (a) - (c) c o n t a i n s a 5-cycle and hence G c o n t a i n s a 5-cycle. In the case (d) , l e t H k - 1 ^ 1 ' A 2 ^ — where ^ = [ x - ^ X g , . . . > ^ L _ 1 ) and Ag = { y 1 , y 2 , • • • ,y k_ x} • Then ( x - L x 2 ) h 1 , ( x 2 x ^ ) h g , ( y - L y 2 ) h ^ € G f o r some t r a n s -p o s i t i o n h± e Sym({a ,0,Y}) , i = 1 , 2 ,3 . I f h^ = h 2 , then (x-jXgX^) = ( x 2 x ^ ) h 2 ( x 1 x 2 ) h 1 e G . I f + h g , then we may assume ^ y and t h e r e f o r e the square of ( x 1 X g ) ( y ] _ y 2 ) n i n 2 i s a 3-cycle i n G . Hence A l t (N) c G . T h i s completes the p r o o f . We summarize the r e s u l t s o f Theorem 4.3, C o r o l l o r y 4.6 and Theorem 4.18 i n Theorem 4 . 1 9 - Let G be a (k)-group of degree n <_ 3k - 1 , k _> 4. . \ Then G c o n t a i n s a subgroup o f type ( i ) S k + 1 , ( i i ) A ^ g , ( i i i ) S k + 1 * S k + 1 or ( i v ) H k . / Lemma 4.20. L e t G c G-^  x. Gg be such t h a t the r e s t r i c t i o n map ^ : G G^, i s onto, i = 1 ,2 . Then f g : G-j/ker (j)g = Gg/ker (j^ and G = ( G - ^ k e r ^ g , ^ ) * ( G g j k e r ^ f g ) where f ^ i s the i d e n t i t y map on G-^/ker (j)g . 64. Proof: Let = ker <j)g and Kg = ker (j)^  . .For each -g . l K l € G 1 ^ K 1 3 t i i e r e e x i s ' t s g 2 e Gg such t h a t gjgg e G * and hence we d e f i n e a map f g : G-j/K^ - Gg/Kg by p u t t i n g f 2 ^ g l K l ) = g 2 K 2 f g i s w e l l - d e f i n e d , f o r i f there e x i s t s hg e Gg such t h a t gjhg e G , then ggh" 1 = ( g ^ g ) ( g ^ g ) - 1 e G . T h i s i n t u r n i m p l i e s t h a t gghg"^ € k e r ty^ = Kg and hence ggKg = hgKg . Moreover, i t i s not d i f f i c u l t to check t h a t f g i s one to one and p r e s e r v e s the o p e r a t i o n . Hence f g i s an isomorphism. Next, we show t h a t G = (Gi>K±>?±)* ( G 2 , K 2 , f 2 ) . C l e a r l y G c ( G ^ K p f ^ ) * ( G 2 , K g 3 f 2 ) - On the o t h e r hand, l e t g±g2 e ( G ^ K - ^ f ^ ) * ( G 2 , K 2 , f 2 ) . Then fg(g^K^) = ggKg , and there e x i s t s hg e Gg such t h a t g^hg € G and hence we have f 2 ^ g l K l ^ = n 2 K 2 " Therefore gg G hgKg , and hence g±£>2 e h i h 2 K 2 — G ' Th-113 completes the p r o o f . \ Lemma 4.21. L e t G c x Gg where Gg i s o f type H k 3 S k " ^ A 3 3 St o r ^k 3 a i ^ d k — ^ 3 a n d t h e r e s ' t r i c ' f c i o n map ty^ : G -• G 2 i s onto Then G has a 3-cycle. Proof: Let (J)^ : G -• be the r e s t r i c t i o n map. Let {N^jNg} i n the f i r s t case ({N-^N^N^} i n the remaining cases) be a complete"block system of G 0 . I f K = k e r ^ has no 3-cycle, then |Sym(N iVSyin^)DK| ( r e s p e c t i v e l y , | Sym(N i )*Sym(N . )/Sym(N i )*Sym(N )riK| ) i s 6 5. o d i v i s i b l e by 2 . Therefore |G0/K| i s d i v i s i b l e by 2 . But t h i s i s im p o s s i b l e s i n c e Im (j^/ker <J)0 = Gg/ker <j)-^  and |lm (J^/ker <()2 | i s a f a c t o r o f 6 . Hence K c o n t a i n s a 3 - c y c l e . . ,. I Theorem 4.22. L e t G be a p r i m i t i v e (k)-group o f degree n = 3k where k 3 . Then A l t ( N ) c G . Proof: We use i n d u c t i o n on k to prove t h i s theorem. By Lemma 4.12. I t i s true f o r k = 3 . Now, suppose the theorem i s true f o r a l l p r i m i t i v e (k - 1 )-groups o f degree 3 3 k >. 4 . L e t Q b i a p r i m i t i v e (k)=group o f degree 3k . By Theorem 2 . 15 and Theorem 4 . 1 3 , i t i s s u f f i c i e n t to show t h a t G c o n t a i n s a 3 - c y c l e , a N 5 - c y c l e or a 7 - c y c l e . Suppose t h a t A l t ( N ) G . Let a,p,y be d i s t i n c t elements i n N and l e t A = N - {a,p,v} • Then G A i s a (k - 1 )-group of degree <_ 3 (k -1 ) . I f G A i s p r i m i t i v e on A , then by the i n d u c t i o n hypothesis G A 3 A l t ( A ) and hence A l t ( N ) c G , a c o n t r a d i c t i o n . I f G A i s not p r i m i t i v e on A , then, by Theorem 4.10 , Lemma-4.16, Lemma 4.17 and Theorem 4 . 1 9 , G c o n t a i n s a subgroup o f type ( i ) S f c , ( i i ) S k * S k , ; ( i i i ) A k + 1 , ( i v ) S 2 ^ S 4 ( I f k = 4 ) , (y) PSL(2 , 7 ) ( i f k = 4) , ( v i ) , ( v i i ) S k _ ; | y A 5 , ( v i i i ) Q k_ 1 or ( i x ) ^ k_ ]_ . For the cases ( i i i ) and (v) , we can e a s i l y show t h a t "G has a 5 - c y c l e o r a 7 - c y c l e . F o r the cases 66. ( v i ) - ( i x ) G has a 3 - c y c l e by Lemma 4 . 2 1 . The .proof o f the case ( i v ) i s p r e c i s e l y the same as t h a t o f Theorem 4.18. Therefore i t remains to c o n s i d e r the case ( i ) and the case ( i i ) . Case ( i ) I f k > 5 , t h i s reduces to case ( i i i ) . \ Hence we need o n l y to c o n s i d e r the case k = 4 . Let Sym(A 1) c H where & = { 1 ,2 ,3 ,4} . Then C = { ( 1 2 ) ( 3 4 ) , ( 1 3 ) ( 2 4 ) , ( l 4 ) ( 3 2 ) , ( 1 2)(aB ) , ( 3 4)(ap)} c G . Since G i s p r i m i t i v e , there e x i s t s g e G such t h a t a e g(A^) and B £ &(^) • I f fiC^) n b±. = ty , then the square o f ( 1 2)(aB)g - 1 ( 1 2 ) ( 3 4)g i s e i t h e r a 3 - c y c l e o r a 5 - c y c l e . I f |g()HA-^| = 1,3 , then a g a i n the square of ( 1 2 ) ( 3 4)g " 1 ( 1 2 ) ( 3 4)g i s e i t h e r a 3 - c y c l e or a 5 - c y c l e . I f |g(A-^)HA-^ | = 2 , then C U g - ± C g generates a t r a n s i t i v e group on g ( A 1 ) U A-L U {6} o f degree 7 . Therefore the order of < C U g - 1 c g > i s d i v i s i b l e by 7 and hence i t co n t a i n s a 7 - c y c l e . T h i s i n t u r n i m p l i e s t h a t G has a 7 - c y c l e . \ Case ( i i ) I f k j> 5 , then H c o n t a i n s a 5 - c y c l e and hence G has a 5-cycle/. Now, suppose t h a t k = 4 . L e t Sym(A 1)*Sym(A 2) c H where A 1 = { 1 , 2 , 3 , 4 } and ^2 = ( 5 , 6 , 7 , 8 } . Then we may assume t h a t (123)(aBv) , (567)(aBy) , ( 12) (56)(aB) e G', and t h e r e f o r e (134)(aBy) = ( 1 2 ) ( 3 4 ) ( 1 2 3)(ap Y) and ( 5 7 8)(aBy)= ( 5 6 ) ( 7 8 ) ( 5 6 7 )(aB Y) are i n G Let A = ' {1 ,2, . . . ,9} • - Consider a p a r t i t i o n P = {XlfX2,Xy\} on N d e f i n e d by X± = { 1 , 2 , 5 , y } , X 2 = ( 6 , 7 , 3 } , = {4 ,8 ,9} , X 4 = {a,B} . L e t g be a 6 7 -P - e l e m e n t i n G . Then we have the f o l l o w i n g c a s e s : ( a ) One o f t he s e t g ( [ l , 2 , 3 } ) n { 5 , 6 , 7 } , g ( { 5 , 6 , 7 } ) n { 1 , 2 , 3 } i s n o t empty . I n t h i s c a s e , < ( 1 2 3 ) ( 5 7 6 ) , g - 1 ( 1 2 3 ) ( 5 7 6 ) g > i s a p r i m i t i v e g roup o f degree 7 i f y i s n o t f i x e d by g , and < ( 1 2 3 ) ( 5 7 6 ) , g - 1 ( 1 2 3 ) ( 5 7 6 ) g > i s a p r i m i t i v e g roup o f degree 6 i f y i s f i x e d by g . I n t h e f i r s t case G has a 7 - c y c l e . I n t h e second case G has a 5 - c y c l e b y Lemma 2 . 9 . (b ) g ( { l , 2 , 3 } ) fl { 5 , 6 , 7 } = g ( { 5 , 6 , 7 } ) 0 { 1 , 2 , 3 } = <j) and Y i s n o t f i x e d by g . I n t h i s c a s e , we may assume t h a t g ( l ) = Y A N D g ( Y ) = { 1 , 2 } . Then < ( 1 2 3 ) ( a P Y ) , g - 1 ( 1 2 3 ) ( a P Y ) g > i s a p r i m i t i v e group. 1 o f degree 6 , and hence G c o n t a i n s a 5 - c y c l e . ( c ) g ( { l , 2 , 3 } ) fl { 5 , 6 , 7 } = g ( { 5 , 6 , 7 } ) fl { 1 , 2 , 3 } and Y i s f i x e d b y g . I n t h i s c a s e , g = g-jggg^ where g± e { 1 , ( 1 2 ) } , S 2 6 { 1 > ( 6 7 ) } , g-j e \ { 1 , (a0)} and g^ i s i d e n t i t y o r a t r a n s p o s i t i o n i n S y m ( { 4 , 8 , 9 } ) • I f g ^ = 1 , t h e n e i t h e r ( 1 2 ) ( 5 6 ) ( a p ) g i s a t r a n s p o s i t i o n o r i t s square i s a 3 - c y c l e . So we may assume t h a t g^ 4 1 • W e obse rve t h a t i f g 4 = (48) , t h e n < ( 2 1 4 ) ( 5 6 8 ) , g _ 1 ( 2 l 4 ) ( 5 6 8 ) g > i s a p r i m i t i v e g roup o f degree 6 c o n t a i n e d i n G , s i n c e ( 2 1 4 ) ( 5 6 8 ) = ( 3 2 ) ( 1 4 ) . ( 1 2 3 ) ( 5 7 6 ) - ( 6 7 ) ( 5 8 ) i s i n G . Hence b y Lemma 2 . 9 G has a 5 - c y c l e . On the o t h e r h a n d , i f g = (49) o r g = (89) , t h e n e i t h e r A l t ( A 1 U { 9 } ) . £ GA o r A l t ( A 2 U { 9 } ) c GA , i . e . G A c o n t a i n s a subgroup o f t y p e A . T h i s reduces t o case ( i i i ) ... Th i s comp le tes t he p r o o f . 68. We summarize a l l the above r e s u l t s i n the f o l l o w i n g Table. n k Types o f minimal (k)-groups c o n t a i n e d i n a (k)-group of degree n < 2k 1  2 Sk+1 > \ + 2 • = 2k >.  2 Sk+1 3 \ + 2 3 H k * = 2k + 1 = 2 = 3 S 4 , , , PSL ( 2 , 7 ) -> 4 S k + l ' ' Ak+2 3 H k ' = 2k + 2 = 3 s 4 * A 5 v H 3 > V S 4 > P S L ( 2 > ? ) , S 2"VS 4 . j > 4 Sk+T ' Ak+ 2 > H k 3 S k + l * S k + l • 2k+2<n<3k > 4 • Sk+1 3 \ + 2 ^ H k 3 S k + l * S k + l • = 3k = 2 s3 . > \ > H 2 * D 5 * S 2 ' V A 3 > S2 > S? ' S 3 * S 3 = 3 S 4 , A 5 , S U * S 4 , , S \ A , Q '5 Q , S 2~V.S 4 , PSL ( 2 , 7 ) • • _> 4 Sk+1 3 \ + 2 > S k + l * S k + l * H k 3 S k ^ A 3 > \ 3 \ • . .; 69-Chapter V Non-primitive (k)-groups of degree n , 3k < n <^  4k . In Chapter I I I , we have proved that a=l*( G±>K^>§^) i s a minimal (k)-group whenever i s a (k)-group. We s h a l l obtain a l l the i n t r a n s i t i v e minimal (k)-groups of degree n <_ 4k by applying this f a c t . § 1 . Intransitive minimal (k)-groups of degree n , 3 k < n ± 4k . Lemma 5«1> Let G be a minimal i n t r a n s i t i v e (k)-group of degree 3k + 1 .." Then G has only two orbi t s ^ and N 2 with IN-J = k + 1 and |N2| = 2k . Proof: I t i s clea r that G has two orbits N, and \ N, 1 N p N 2 with IN-J >_ k + 1 , |N2| >_ k + 1 , and G and G a r e (k)-groups. For k = 2 , n = 7 , i t i s obvious. Now suppose that 2k > |Nn | > k + 2 and 2k > |Np| > k + 2 N. for some k > ) . Then by Corollory 2.6 Alt(N,) c G 1 ' N. for i = 1,2 . Since the r e s t r i c t i o n map f : G -» G 1 i s onto for i = 1 ,2 , , by Lemma 4 .20 we have (j) : N , N p N-, G Vker f p ~ G V k e r f and G = (G ,ker fp,(J), )* N 2 ^ " 1 d 1 N, (G ,ker f , V 4 P ) where (j) i s the i d e n t i t y map on G /ker f - N N Note that i f ker f, = {1} = ker f p .,. then G 1 = G 2 , 70. a n d t h e r e f o r e G i s c o n j u g a t e t o a g r o u p d e f i n e d i n T h e o r e m 3 - H o r T h e o r e m 3 . 1 2 . B u t , s i n c e t h e d e g r e e o f G i s l e s s t h a n 2 ( k + 1 ) , G i s n o t a ( k ) - g r o u p , a n d t h i s c o n t r a d i c t s t h e a s s u m p t i o n o n G . H e n c e we m u s t h a v e k e r f 1 4 {1} o r k e r f 0 4 {1} , s a y k e r f 1 4 {1} • S i n c e k _> 3 , a n y n o n - t r i v i a l n o r m a l s u b g r o u p o f s k + 2 o r ' A k + £ m u s t c o n t a i n Ak + 2 • T h u s we h a v e k e r f ^ 3 A l t ( N ^ ) , a n d s i n c e A l t ( N ^ ) i s a k - g r o u p , t h i s c o n t r a d i e t s t h e m i n i m a l i t y o f G . H e n c e | N-J. = k + 1 a n d | N 2 | = 2 k . Lemma 5.2. L e t G b e a n i n t r a n s i t i v e m i n i m a l (2)-g r o u p o f d e g r e e 7 . T h e n G i s e i t h e r o f t y p e T = < (12) (56) , (3*0(56) , (15) (67) , ( 2 4 ) (67) > o r ( S 3 , A 3 , f 1 ) * ( H 2 , K 1 , f 2 ) w h e r e K± e ^ 2 ( H 2 ) , f1 i s t h e i d e n t i t y map o n S - j / A ^ a n d f 2 ' ^-^/A-^ -* H 2 / K - ^ i s t h e o b v i o u s i s o m o r p h i s m . P r o o f : L e t = {(12) (34), (13 ) . ( 2 4 ) , ( 1 4 ) (23), 1 } . T h e n S 4 / V 4 ~ S ^ , a n d ( S 4 , V 4 , 1 ) * ( S ^ ( l } , ( j ) ) a n d T a r e o f t h e s a m e t y p e . Now l e t N , a n d N P b e t w o , N o r b i t s o f G w i t h | N , | = 3 a n d | N 0 | = 4 . T h e n G -1 N 112 i s o f t y p e S a n d G i s o f t y p e Sk , A . o r H ? . 1 N B y Lemma 4.20 we h a v e G = ( S , y K , , f , ) * ( G 2 , K 0 , f 0 ) . I n N • t h e c a s e G = S 4 we h a v e K 1 = {1} , K 0 = V" 4 o r ~ A ^ , K 2 = A 4 o r ^ = S ^ , K 2 = S 4 . T h e f a c t s 7 1 . t h a t K g c G a n d a n d . a r e ( 2 ) - g r o u p s , t o g e t h e r w i t h t h e m i n i m a l i t y o f G e n a b l e u s t o c o n s i d e r t h e f i r s t c a s e o n l y . T h e n G i s o f t y p e T . I n t h e c a s e N 2 ~ G = A ^ , we h a v e K g = A ^ , b e c a u s e t h e o n l y n o n -t r i v i a l p r o p e r n o r m a l s u b g r o u p o f A h i s V k o f i n d e x N x f g N g ^ 4 3 , a n d G / K , ~ G / K p . T h i s c o n t r a d i c t s t h e m i n i m a l i t y N 2 ~ o f G . F i n a l l y , i n t h e c a s e G = H g , T h e o r e m 3 . 8 i m p l i e s t h a t | S ^ / K - ^ | = | H g / K g | = P , a p r i m e . T h e r e f o r e P ^ 2 a n d t h i s c o m p l e t e s t h e p r o o f . T h e o r e m 5 . 3 . L e t G b e a m i n i m a l i n t r a n s i t i v e ( k ) - g r o u p o f d e g r e e n =. 3 k + 1 , k _> 3 • T h e n G i s o f t y p e ( S k + 1 J A k + 1 , f 1 ) * ( H k , K , f g ) w h e r e K e ff2(Hk) . P r o o f : ' T h e p r o o f i s s i m i l a r . to t h a t o f L e m m a 5 . 2 . ' L e t N , , N ? b e o r b i t s o f G w i t h | N , | = k + 1 , N , N INpI = 2 k . T h e n , b y L e m m a 4 . 2 0 , G = ( G 1 , K 1 , f 1 ) * ( G , K 0 , f 0 ) N , N p . N-. Np w h e r e f p : G / K , = G / K p . S i n c e G a n d G a r e % d Np ( k ) - g r o u p s , G i s o f t y p e S, , a n d G i s o f t y p e , K + 1 N 2 H k ' A 2 k 0 r S 2 k ' I n t h e c a s e G z2 A l t ( N g ) , we h a v e G z> A l t ( N p ) , w h i c h c o n t r a d i c t s t h e m i n i m a l i t y o f G . N g T h e r e f o r e G i s o f t y p e H k . N o w , s i n c e t h e i n d e x o f a n o r m a l s u b g r o u p o f S, , h a s a f a c t o r 2 , T h e o r e m N + N 3 - 8 i m p l i e s t h a t | G VK-J = | G 2 / K g | h a s t o b e e q u a l t o 2 . T h i s c o m p l e t e s t h e p r o o f . 7 2 . T h e o r e m 5 . 4 . . I f 3 k + 2 <_ n < 4 k , k > 3 , a n d i f G i s a ( k ) - g r o u p o f d e g r e e n w i t h o n l y t w o o r b i t s N ^ a n d N 2 , t h e n G i s n o t m i n i m a l . N, P r o o f : • B y L e m m a 4 . 1 , we m a y s u p p o s e t h a t G a n d N N 1 G a r e ( k ) - g r o u p s , a n d b y L e m m a 4 . 2 0 G = ( G , K n , f - , ) * N ? 1 N t ( G , K 2 , f 2 ) . A s s u m e |N | < 2 k . T h e n A l t ( N 1 ) c G a n d we h a v e t h e f o l l o w i n g c a s e s : N N p p C a s e 1 . G i s p r i m i t i v e . T h e n G z> A l t ( N Q ) o r N ~ G 2 i s o f t y p e P S L ( 2 , 7 ) ( i f k = 3 , n = 1 1 a n d = 4 S i n c e P S L ( 2 , 7 ) i s s i m p l e ( [ 1 ] . , p . - 2 1 7 ) a n d N, N p ! N, Np G V K , ~ G V K p , we m u s t h a v e 'G = K-, a n d G = K Q . N 2 • - 1 ' d T h e r e f o r e i f G = E t h e n G c a n n o t b e m i n i m a l . O n N t h e o t h e r h a n d i f A l t ( N ) c G , t h e n b y L e m m a 4 . 2 we h a v e A l t ( N p ) c G , a n d G i s a g a i n n o t m i n i m a l . N / 1 N 2 C a s e 2 . G i s n o t p r i m i t i v e . L e t { A - , , A ^ } b e t h e N 2 l a , c o m p l e t e b l o c k s y s t e m o f G . T h e n b y C o r o l l o r y 4 . 5 , N A l t ( A i ) c G f o r ±\ = 1 , 2 . S i n c e k _> 4 i n t h i s c a s e , A l t ( N ^ ) i s s i m p l e . T h e r e f o r e = { 1 } o r c o n t a i n s A l t ( N , ) . I f K-, i s n o t a ( k ) - g r o u p , t h e n K , = { 1 } 1 x N , o r i s o f t y p e A^ . ^ ( i n w h i c h c a s e G = S y m ( N ^ ) ) . H e n c e t h e r e e x i s t s a p a r t i t i o n P ^ = { X - ^ X g , . . .,X^} o n N ^ with | X i | <_ 2 s u c h t h a t n o P ^ - e l e m e n t i s i n . L e t B , b e t h e s e t c o n s i s t i n g ' - o f a l l t h e P , - e l e m e n t N, . 1 t o g e t h e r w i t h t h e i d e n t i t y i n G . T h e n B 1 fl K ^ . = [ 1 } a n d t h e o r d e r o f t h e g r o u p B-, i s a p o w e r o f 2 . We s h o w N , t h a t t h e g r o u p B = [ g 1 g 2 e G |. g 1 e B^^ a n d g £ e G } c o n t a i n s A l t ( A . ) , i = 1,2 . T o d o t h i s , we f i r s t s h o w 1 N2 t h a t B 0 = [ g 2 e G | S g 1 e B- L w i t h g 1 g 2 e G} i s a ( k ) - g r o u p . I n f a c t , f o r e a c h p a r t i t i o n ' 1 4 P 2 = { Y ^ , Y 0 , . . , Y ^ } o n N 2 , P = { X 1 U Y 1 , . . . , X k U Y k ) i s a p a r t i t i o n o n N . T h e n b y t h e a s s u m p t i o n o f G , t h e r e i s a P - e l e m e n t N N 2  S1S2 € G w h e r e § i e G a n d 6 2 6 G • C l e a r l y ^ 1 H e n c e B 2 i s a ( k ) - g r o u p o n N 2 . T h e r e f o r e b y T h e o r e m 4.19, B 0 c o n t a i n s a s u b g r o u p o f t y p e A k . T h i s i n t u r n i m p l i e s t h a t G c o n t a i n s a s u b g r o u p o f t y p e A , . N K T h e n , b y t h e t r a n s i t i v i t y o f G , G c o n t a i n s A l t ( A . ) , i = 1,2 , i . e . K r> A l t ( A . ) , i - 1,2 . N o w , ' 1 N , N N 1 s i n c e G 1/K-L = G 2 / K 0 a n d | G 2 / K 2 | i s a p o w e r o f 2 , we m u s t h a v e K 1 3 A l t ( N 1 ) a n d IN.J = k + 1 . T h u s I A j J _> k + 1 . I n t h e c a s e | A ^ | >_ k + 2 , b e c a u s e o f t h e f a c t t h a t A l t ( A . ) i s a k - g r o u p , we a r e d o n e . O t h e r -1 N w i s e , b y C o r o l l o r y 4.5, S y m ( A 1 ) * S y m ( A 2 ) c N 2 . I f S y m ( A n ) * S y m ( A p ) <=• K p , t h e n G i s n o t m i n i m a l . I f N S y m ( A 1 ) * S y m ( A 2 ) £ K o V , t h e n , | G 2 / K 2 | = | S y m ) / A l t ) | = i m p l i e s t h a t G p r o p e r l y c o n t a i n s S y m ( N 1 ) * S y m ( A - L ) * S y m ( A 2 ) w h i c h i s a k - g r o u p . H e n c e ,-G i s n o t m i n i m a l . T h i s c o m p l e t e s t h e p r o o f . L e m m a 5.5. L e t G c G-^ x G 2 x b e s u c h t h a t G ^ <{:.G f o r a l l i , a n d s u p p o s e t h a t t h e r e s t r i c t i o n m a p s ( j ) i : G - » G i , i = l,2,3 a r e o n t o . L e t b e a m a x i m a l s u b g r o u p o f G ^ c o n t a i n e d i n G , i = 1,2,3 • 74. I f K± 6 ! J p ( G i ) , i = 1 , 2 , 3 , f o r some p r i m e P , t h e n G = ( ( G 1 , K 1 , f 1 ) » ( G 2 , K 2 , f 2 ) ) ( ( G 1 , K 1 J f 1 ) * ( G 5 , . K 5 , f 3 ) ) o r G = i l i * ( G i * K i , f . ) f o r s o m e f . : G 1 / K 1 = G . / K . , i = 1 , 2 , 3 • P r o o f : L e t K Q b e t h e k e r n e l o f t h e r e s t r i c t i o n m a p (J)^ : G -» G " a n d l e t G Q b e t h e i m a g e o f t h e r e s t r i c t i o n m a p ^ : G G ^ x G g . ' N o t e t h a t k e r ^ = K -a n d h e n c e b y L e m m a 4 . 2 0 , G / K ' = G V / K . , i s o f o r d e r P . J ' o o 3 3 We o b s e r v e t h a t K => K , x K 0 . I f K = K , x K 0 , t h e n o — l 2 o 1 2 * G Q = ( G 1 , K 1 , f 1 ) * ( G 2 , K 2 , f 2 ) f o r s o m e f 2 : G - j / K ^ = G 2 / K g , a n d h e n c e G = * ( G . , K . , f . ) f o r s o m e f . : G . / K . = G./K. i = l v i * i * i y I 1 1 i i i = 1 ^ 2 , 3 . O n t h e o t h e r h a n d , i f K q + x K 2 , t h e n G-^ x G 2 ch G a n d t h e f a c t t h a t G ^ / K ^ i s o f p r i m e o r d e r i m p l y t h a t ( G 1 , K 1 , f 1 ) * ( G 2 , K 2 , f 2 ) = K Q C G f o r s o m e f 2 : G 1 / K 1 = G 2 / K 2 . S i m i l a r l y , ( G ^ K - ^ f . ^ * ( G - ^ K ^ f ^ ) c G f o r s o m e f : G 1 / G 1 = G ^ / K ^ . P u t H = ( ( G 1 , K 1 , f ] L ) * ( G 2 , K 2 , f 2 ) ) ( ( G 1 , K 1 , f 1 ) * ( G 5 , K ^ , f 5 ) ) . S i n c e KQ X i s c o n t a i n e d i n H a n d t h e i n d e x e s o f K i n H a n d i n G a r e t h e s a m e , we h a v e / H = G . T h i s c o m p l e t e s t h e p r o o f . L e m m a 5 . 6 . L e t G b e a n i n t r a n s i t i v e m i n i m a l ( 3 ) -g r o u p o f d e g r e e 1 2 . T h e n G i s o f t y p e S 4 * S 4 * S 4 , ( S j 4 , A 4 , f 1 ) * ( S 2 x S 2 f , K 2 , f 2 ) - o r , ( H 3 , K 3 , f 3 ) * ( H 3 , K 4 , f 4 ) , w h e r e K 2 e J ^ S ^ ) , , K 4 e a n d f - ^ f ^ f ^ a r e t h e o b v i o u s i s o r m o r p h i s m s . P r o o f : F i r s t we s u p p o s e t h a t G h a s t w o o r b i t s N , N , a n d N 9 w i t h | N , | < 6 a n d | N J > 6 . T h e n G a n d G a r e ( 3 ) - g r o u p s a n d G i s a s t a r p r o d u c t N , N o f G a n d G . N o t e t h a t i f |N- . | = 5 , t h e n N N 1 A l t ( N - , ) c G a n d G i s p r i m i t i v e , a n d t h e r e f o r e N N A l t ( N ) c G 2 o r G 2 i s o f t y p e P S L ( 2 , 7 ) . S i n c e A l t ( N 2 ) a n d P S L ( 2 , 7 ) a r e s i m p l e , b y L e m m a 4.20 , i t f o l l o w s t h a t A l t ( N 0 ) c G o r G c o n t a i n s a s u b g r o u p o f t y p e P S L ( 2 , 7 ) w h i c h c o n t r a d i c t s t h e m i n i m a l i t y o f G . . T h e r e f o r e | N , | = 4 o r | N | = 6 . 1 % , C f l s e l . | N J = 4 . T h e n G = S y m ( N , ) . I f G i s p r i m i t i v e , t h e n b y t h e s a m e a r g u m e n t a s a b o v e , G i s n o t m i n i m a l . H e n c e , b y C o r o l l o r y 4.5 a n d C o r o l l o r y 4.16, N 2 N G i s o f t y p e S 0 ~ \ S ^ o r G c o n t a i n s a s u b g r o u p o f t y p e S 4 * S 4 . N o w , s i n c e S y m ( N - L ) cj; G , we m a y c h o o s e a p a r t i t i o n P 1 = { X - ^ X ^ X - ^ } o n N ^ s u c h t h a t t h e o n l y P ^ - e l e m e n t i n S y m ( N ^ ) i s a t r a n s p o s i t i o n g w h i c h i s n o t i n G . T h e n , f o r e a c h p a r t i t i o n P 2 = { Y ^ Y ^ Y ^ } o n . N g , P = { X 1 U Y 1 , X 2 U Y 2 , X ^ . U Y 5 } i s a p a r t i t i o n o n N . B y h y p o t h e s i s , t h e r e e x i s t s a P - e l e m e n t g i n G . We o b s e r v e t h a t t h e r e s t r i c t i o n o f g o n N 0 i s n o t t r i v i a l . N T h e r e f o r e B = {g e G 2 | g e G o r g g e G} i s a d N | 1 d ( 3 ) - g r o u p c o n t a i n e d i n G , a n d h e n c e B i s o f t y p e S 2 ^ S 4 o r B c o n t a i n s a s u b g r o u p o f t y p e S ^ * S 4 . T h i s i m p l i e s t h a t G c o n t a i n s a s u b g r o u p o f t y p e K 2 o r A ^ x A ^ w h e r e K g e 5 2 ( S 2 " \ S 4 ) . B y L e m m a 4.20 , t h e f i r s t c a s e s h o w s t h a t G i s o f t y p e ( S ^ A ^ , f 1 ) * ( S 2 \ S ^ , K 2 J f 2 7 6 . a n d t h e l a t t e r c a s e s h o w s t h a t G p r o p e r l y c o n t a i n s a s u b g r o u p o f t y p e S 4 * S 4 * S 4 , w h i c h c o n t r a d i c t s t h e « (i; m i n i m a l i t y o f G . N , N C a s e 2 . I N - J = 6 . I f G 3 A l t (N-^) a n d G 3 A l t ( N 2 ) , t h e n b y L e m m a 4 . 2 0 , T h e o r e m 3 - 1 1 a n d T h e o r e m 3 - 1 2 , G h a s t o c o n t a i n A l t ( I \ L ) a n d A l t ( N ^ ) , a c o n t r a d i c t i o n . 1 N . ^ N , S u p p o s e t h a t o n e o f G 1 , i = 1 , 2 , s a y G , c o n t a i n s N N • N A l t ( N , ) a n d G 2 i s o f t y p e H . T h e n | G 2 | < | G 1 | . H e n c e G c o n t a i n s a n o n - t r i v i a l n o r m a l s u b g r o u p o f G b y L e m m a 4 . 2 0 . T h i s n o r m a l s u b g r o u p m u s t b e A l t ( N ^ ) , w h i c h a g a i n c o n t r a d i c t s t h e m i n i m a l i t y o f G . T h e r e f o r e N l N 2 b o t h G a n d G a r e o f t y p e H ^ a n d h e n c e b y L e m m a 4 . 2 0 a n d T h e o r e m 3 - 8 G i s o f t y p e ( H ^ K ^ , f ^) * ( H ^ K ^ , f ^) f o r s o m e , K 4 e ^ ( H ^ ) . O n t h e o t h e r h a n d . , s u p p o s e t h a t G h a s 3 o r b i t s N i N 3 , N 2 a n d . T h e n G .= S y m ( N 1 ) f o r i = 1 , 2 , 3 . S i n c e S y m ( N 1 ) * S y r a ( N 2 ) ch G , ( x - ^ H y ^ ) ^ G f o r s o m e , x 2 e N-^ a n d y ^ , y 2 e N 2 , a n d t h e r e f o r e we c a n c h o o s e a n a p p r o p r i a t e p a r t i t i o n P-^ = {X-^Xp^X^} o n N , U N 2 N-^ U N 2 s u c h t h a t e v e r y P - ^ - e l e m e n t i n G i s a f a c t o r o f ( x - ^ x 2 ) ( y ^ y 2 ) . T h e n b y t h e s a m e a r g u m e n t a s i n c a s e 1 , a n o n - t r i v i a l n o r m a l o f s u b g r o u p i n S y m ( N 3 ) o f i n d e x a p o w e r o f 2 m u s t b e , c o n t a i n e d i n G ". T h i s n o r m a l s u b g r o u p h a s t o b e A l t ( N ^ ) . S i m i l a r l y , we c a n s h o w t h a t A l t ( N - L ) , A l t ( N 2 ) c G . H e n c e b y L e m m a 5.5 G i s o f t y p e S 4 * S 4 * S _ 4 . T h i s c o m p l e t e s t h e p r o o f . 7 7 T h e o r e m 5 . 7 . L e t G b e a n i n t r a n s i t i v e m i n i m a l ( k ) - g r o u p o f d e g r e e n = 3 k + 3- w h e r e k _> 4 . T h e n G i s o f t y p e \ + 1 * S k + 1 * S k + 1 . P r o o f : B y T h e o r e m 5 . 4 G h a s t h r e e o r b i t s N , , N P N . a n d N , . B y L e m m a 4 . 1 G 1 i s a ( k ) - g r o u p f o r i = 1 , 2 , 3 . ^ N . T h e r e f o r e \N±\ = k + 1 a n d G 1 = S y m ( N i ) f o r i = 1 , 2 , 3 . N o w , l e t  xj_ >  x 2  £ % ^ e s u c n t h a t ( x - ^ X g ) ^ G a n d N - { x , , x ? , y , z } l e t y e N 0 a n d z e N , . T h e n G i s a N - { x 1 , x 2 , y , z } ( k - 1 ) - g r o u p a n d t h e r e f o r e b y T h e o r e m 4 . 3 G c o n t a i n s a s u b g r o u p o f t y p e x A ^ . C l e a r l y t h i s s u b g r o u p h a s t o b e A l t ( N g - { y } ) x A l t ( N ^ - { z } ) . I t f o l l o w s t h a t A l t ( N 2 ) c G a n d A l t ( N ^ ) c G .. S i m i l a r l y , A l t ( N 1 ) c G T h e n b y L e m m a 5 . 5 G i s o f t y p e S k + i * S k + i * S f c + i • L e m m a 5 . 8 . L e t G b e a m i n i m a l i n t r a n s i t i v e ( 2 ) - g r o u p o f d e g r e e 8 . T h e n G i s o f t y p e ( S ^ A ^ , ^ ) * (D , K 2 , f 2 ) , ( H 2 , K 5 , f 5 ) * ( H 2 , K 4 , f 4 ) o r ( A 4 , V 4 , f 5 ) * f ( f 4 , V 4 , f 1 ) w h e r e K 2 - € J 2 ( D 5 ) , , K 4 e a : 2 ( H 2 ) a n d V 4 e ^ ( A 4 ) . P r o o f : B y L e m m a 4 . 1 G h a s t w o o r b i t s N-, a n d N Q , N N . L d a n d G 1 a n d G 2 a r e ( 2 ) - g r o u p s . I f | N | = 3 a n d N , X N j N o I = 5 , t h e n G = S y m ( N , ) a n d e i t h e r G c o n t a i n s N ? 1 A l t ( N 2 ) o r G i s o f t y p e D . B y L e m m a 4 . 2 a n d t h e N m i n i m a l i t y o f G we s e e t h a t G c a n n o t b e A l t ( N ) o r N 2 • S y m ( N g ) . T h e r e f o r e G i s o f t y p e D a n d h e n c e b y 7 8 . L e m m a 4 . 2 0 a n d T h e o r e m 3 - 8 , G i s o f t y p e ( S ^ , A ^ , f 1 ) * ( D ^ , K 2 , f 2 ) • O n t h e o t h e r h a n d , i f N , IN -J = | N 2 | - 4 , we c a n n o t h a v e G z> A l t ( N 1 ) , a n d N 2 N 2 G i s o f t y p e H 2 . F o r o t h e r w i s e , t h e o r d e r o f G N l i s 8 a n d i s l e s s t h a t t h a t o f G . H e n c e b y L e m m a 4 . 2 0 we m u s t h a v e A l t ( N , ) c G , w h i c h c o n t r a d i c t s t h e N m i n i m a l i t y o f G . T h u s we m a y a s s u m e t h a t A l t ( N ) c G N p N N 2 a n d A l t ( N 2 ) c G , o r G a n d G a r e b o t h o f t y p e N N . . ^ H 2 . I f A l t ( N 1 ) c G a n d A l t ( N 2 ) c G t h e n , b y L e m m a 4 . 2 0 , G i s o f t y p e ( A ^ , V ^ , f ) * ( A ^ , V \ , f ^ ) , ( A v { l } , f 5 ) * ( A 4 , { l } , f 6 ) , ( A i f , A l f , f 5 ) * ( A 1 | , A 4 , f 6 ) , ( S 4 , A 4 . , f 5 ) * ( S 4 , A 4 , f 6 ) , ( S 4 , S 4 , f 5 ) * ( S 4 , S 4 , f 6 ) , ( S 4 , V 4 , f 5 ) * ( S 4 , V 4 / f 6 ) o r ( S 4 , { l } , f 5 ) * ( s 4 , { l } , f 6 ) . S i n c e A 4 a n d S 4 . a r e ( 2 ) - g r o u p s , a n d ( A 4 , { l ] , f ^ ) * ( A 4 , ( l } , f 6 ) a n d ( S 4 , { l } , f ^ ) ^ - ( S 4 , { l } , f g ) a r e n o t ( 2 ) - g r o u p s b y T h e o r e m 3 . 1 1 a n d T h e o r e m 3 - 1 2 , t h e m i n i m a l i t y o f G s h o w s t h a t o n l y N , N o t h e f i r s t c a s e c a n o c c u r . F i n a l l y , G a n d G ^ a r e b o t h o f t y p e H p , t h e n b y L e m m a 4 . 2 0 a n d T h e o r e m 3 . 8 we c a n e a s i l y o b t a i n t h a t G i s o f t y p e ( H 2 , K ^ , f ) * ( H 2 , K 4 , f 4 ) w h e r e , K 4 e U 2 ( H 2 ) . L e m m a 5•9• L e t G b e a ( k ) - g r o u p o f d e g r e e n w i t h t h e o r b i t s N , , N „ a n d N_, w h e r e 3 k + 3 < n < 4 k . ± d 3 — T h e n G i s n o t m i n i m a l . " 7 9 -P r o o f : Assume t h e c o n t r a r y , and l e t G be a m i n i m a l ( k ) -group o f d e g r e e n w i t h t h e o r b i t s I\L , N . X N 2 and . By Lemma 4.1 G 1 i s a ( k ) - g r o u p f o r i = 1,2,3 , and s i n c e |N.| < 2k f o r e a c h i , we must N. 1 have A l t ( N . ) cz G 1 f o r i = 1,2,3 • By u s i n g t h e same 1 N 1 U N 2 p r o o f as i n Lemma 4.1 we see t h a t G i s a ( k ) - g r o u p and t h e r e f o r e c o n t a i n s A l t ( N - ^ ) o r A l t ( N 0 ) b y Theorem 4.3 and c o r o l l o r y 2.6. Then b y Lemma 4.20 b o t h A l t ( N ) N-,UN 2 and A l t ( N 2 ) a r e c o n t a i n e d i n G . We s h a l l show t h a t G c o n t a i n s a- s u b g r o u p o f t y p e A 0 and t h e n b y the m i n i m a l i t y o f G we have a c o n t r a d i c t i o n . Assume t h a t |N 1| _> |N 2| > I N ^ I . Then |N 1| _> k + 2 and hence A l t ( N ^ ) c o n t a i n s a s u b g r o u p o f t y p e A^. + 2 . L e t N N n UN.. G = (G p , K 1 , f 1 ) * ( G ^ , K 2 , f 2 ) where ^ c o n t a i n s A l t ( N ^ ) o r i s t h e t r i v i a l s u b g r o u p . I f K 1 3 A l t ( N ^ ) , t h e n K 2 3 A l t ( N 1 ) x A l t ( N 2 ) and hence G c o n t a i n s a su b g r o u p o f t y p e A k + 2 . I f = {1} , t h e n K 2 c o n t a i n s A l t ( N 1 ) o r A l t ( N 2 ) . The f i r s t c a s e shows i m m e d i a t e l y t h a t G has a s u b g r o u p o f t y p e A, . The l a t t e r c a s e i m p l i e s t h a t |N.J = |N 2| '= |N^| > k + 2 and hen c e G c o n t a i n s a s u b g r o u p o f t y p e • T h i s c o m p l e t e s t h e p r o o f . 80 . T h e o r e m 5.10. L e t G b e a m i n i m a l i n t r a n s i t i v e ( k ) - g r o u p o f d e g r e e 4 k , k _> 4 . T h e n G i s o f t y p e ( H k , K 1 , f 1 ) * ( H k , K 2 , f 2 ) w h e r e K± , K g e Z2{\) • P r o o f : B y Lemma 5.9 G h a s o n l y t w o o r b i t s N-^ N N 2 a n d N 2 , a n d G a n d G a r e t r a n s i t i v e ( k ) - g r o u p s . N l N P N o t e t h a t G = (G J - , K 1 , f ] L ) * ( G , K g , f g ) f o r s o m e N-, N p K 3 € V G > J K 2 6 V G } • I f ' N l l < l N 2 ! ^ t h e n N A l t ( N 1 ) c G a n d b y T h e o r e m 4 . 3 , 2 . 2 4 A l t ( A 1 ) , N 2 A l t ( A 2 ) c G , w h e r e { A - ^ j A g } i s a p a r t i t i o n o n Ng . N o w , s u p p o s e K-^ i s n o t a ( k ) - g r o u p , t h e n = {1} . H e n c e t h e r e e x i s t s a p a r t i t i o n P^ = { X - ^ X g , . . . ,X^} o n N-1 w i t h |N.J <_ 2 f o r i = l , 2 , . . . , k , s u c h t h a t n o P - e l e m e n t i s i n ( i . e . n o t i n G ) . L e t B-^ b e t h e s e t o f a l l P ^ - e l e m e n t i n G w i t h i d e n t i t y a n d l e t Bg Ng = ( g 2 € G | 9 &i e B i % € G ^ * W e s h o W t h a t B 2 i S a ( k ) - g r o u p . G i v e n a n y p a r t i t i o n P 0 = f Y , , Y 0 , . . . , Y, } c ± d K. o n N g , P = [ X 1 U Y 1 , X g U Y g , . . . , X k U Y k } i s a p a r t i t i o n o n N T h e n b y t h e a s s u m p t i o n o n G , t h e r e e x i s t s a p - e l e m e n t g l s 2 e G ' c l e a r l y §2 e B 1 a n d 1 4 S 2 e B 2 ' H e n c e B g i s a ( k ) - g r o u p a s r e q u i r e d . T h e n b y T h e o r e m 4 . 1 9 B g c o n t a i n s a s u b g r o u p o f t y p e A ^ ^ • T h i s i n t u r n 8 1 . i m p l i e s t h a t G h a s a s u b g r o u p o f t y p e A k + - ^ • S i n c e G i s t r a n s i t i v e . A l t ( A ^ ) , A l t ( A g ) c G , i.e., N A l t ( A 1 ) , A l t ( A 2 ) c K 2 . T h e r e f o r e G / K g i s o f o r d e r a p o w e r o f 2 , a n d we h a v e A l t ( N ^ ) c K ^ . H o w e v e r , s i n c e n > 3k + 3 , o n e o f A l t ( N ^ ) , A l t (A-^) , A l t ( A 2 ) m u s t c o n t a i n a s u b g r o u p o f t y p e A -^+2 ' a n d t h i s c o n t r a -d i c t s t h e m i n i m a l i t y o f G . T h u s we m a y a s s u m e | N - j J = | N 2 | N . T h e n G 1 i s o f t y p e , o r R" k . N o t e t h a t N . i f A l t ( N i ) c G 1 f o r i = 1,2 , t h e n b y L e m m a 4.20 a n d T h e o r e m 3.11-3.12 we h a v e A l t ( N . ) c G f o r i = 1 , 2 w h i c h N l c o n t r a d i c t s t h e m i n i m a l i t y o f G . A g a i n , i f A l t ( N 1 ) c G N2 a n d G i s o f t y p e , t h e n b y L e m m a 4.20 a n d t h e s i m p l i c i t y o f A l t ( N - ^ ) we h a v e A l t ( N - ^ ) c G , a c o n t r a -N N 2 d i c t i o n . T h e r e f o r e G a n d G a r e b o t h o f t y p e . N , N , , I t f o l l o w s t h e n b y T h e o r e m 3.8, G / K - , a n d G ^ / K „ X d. ^'•^ a r e o f o r d e r p , a p r i m e , a n d h e n c e b o t h a r e a b e l i a n . i • 5 N . T h i s i m p l i e s t h a t K ^ c o n t a i n s t h e d e r i v e d g r o u p o f G 1 f o r i = 1,2 . S i n c e t h e d e r i v e d g r o u p o f H , i s o f i n d e x a p o w e r o f 2 ( w h i c h w i l l b e s h o w n i n S e c t i o n 2 o f C h a p t e r 6), we h a v e p = 2 . T h i s c o m p l e t e s t h e p r o o f . §2 T h e o r e m s o n i m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p s . 8 2 . T h e o r e m 5.11. L e t G b e a n i m p r i m i t v e t r a n s i t i v e ( k ) - g r o u p w i t h c o m p l e t e b l o c k s y s t e m { N ^ , N 2 , . . . , 1 ^ } . A s s u m e e a c h h a s a t l e a s t t + 1 e l e m e n t s . I f k N T s <. ( t ) , t _> 2 , t h e n G i s a ( t ) - g r o u p f o r a l l N i P r o o f : S u p p o s e G i s n o t a ( t ) - g r o u p f o r s o m e N . i . T h e n b y t h e t r a n s i t i v i t y o f G , n o G J i s a ( t ) - g r o u p . F o r e a c h i , l e t P. = [ X . X . ,} b e 1 X ^  X 1 5 XJ a p a r t i t i o n o f l e n g t h t o n s u c h t h a t n o P ^ - e l e m e n t N . i s i n G 1 . L e t B = {X . . j i = l , . . . , s , j = 1 , 2 , . . . , t .} ^ i»re3 r> a n f w i t h t h e p r o p e r t i e s t h a t f(P ) i s a s u b s e t o f t S i n c e s < (^) , we c a n f i n d a f u n c t i o n f : B - [ l , 2 , . . . , k ] e l e m e n t s i n { l , 2 , . . . , k } a n d f o r e a c h p a i r i a n d io , i , 4 io , f(P. 0 4 f(P- ) • N o w , we c o n s t r u c t X c. X X ^ a . p a r t i t i o n P = { Y 1 , . . . , Y k } o n N a s f o l l o w s : Y = U X m = 1 , 2 , . . . , k m f ( X . . )=m l j J 1 > J N o t e t h a t f o r e v e r y p a i r i ^ a n d i 2 , i ^ 4 ^2 3 i s m e { l , 2 , . . . , k } s u c h t h a t m e f ( P . ) b u t 1 1 ni I f ( P . ) . T h u s t h e r e e x i s t s Y m e P w i t h t h e r e Y • fl N . 4 ty a n d Y n N . = ty . T h i s m e a n s t h a t e v e r y ni x -j^  m x g 8 3 -P - e l e m e n t g i n G m u s t l e a v e e v e r y b l o c k i n v a r i a n t . i . e - g ( N ± ) = N i f o r a l l i . ' T h e r e f o r e g | N ± 4 1 f o r s o m e i , w h i c h c o n t r a d i c t s t h e a s s u m p t i o n o n P.. N . H e n c e G i s a ( t ) - g r o u p f o r a l l i . T h e o r e m 5 . 1 2 . L e t G b e a n i m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p w i t h c o m p l e t e b l o c k s y s t e m {N-, , N 0 , . . . , N } , s u c h t h a t I N ^ J > t + 1 f o r a l l i . I f | N ± | ^ 0 (mod t ) f o r a l l i a n d s <_ t . (^) , t h e n G i s a ( t ) - g r o u p f o r a l l i . P r o o f : A s s u m e t h e c o n t r a r y . L e t P.. = [ X ^ ^ , . . . , X ^ ^} b e a p a r t i t i o n o f l e n g t h t o n N ^ s u c h t h a t n o P.. -e l e m e n t i s i n G ^ , i = l , 2 , . . . , s . W i t h o u t l o s s o f g e n e r a l i t y , we m a y a s s u m e |x . , | > | X . Q \ > . . . > J X . , | f o r a l l i . N o w , f o r a n y t p a r t i t i o n s ( o r l e s s t h a n t " p a r t i t i o n s ) i n { P . | i = l , 2 , . . . , s ] , s a y P , , P 0 , . . . , P , , w e ' d e f i n e a p a r t i t i o n P = { Y - ^ Y ^ . . . jY^. } o n U N 2 U . . . U N t a s f o l l o w s : Y l = X l , l U X 2 , 2 U ••• U X t , t Y 2 = X l j 2 U X 2 ^ 3 U . . . U X t j l Y t = X l , t U X 2 , l U . . . . U Xt)t_1 84. Since |N. | ^ 0 (mod t) , there are X. .. and X.. . with JX . I 4 | X . • I f o r each i . Therefore f o r ^ 3 -L J J g d i s t i n c t 1-^,1^ e { 1 , 2 , . . . , t } there e x i s t s Y^ e P such that lY^ fl N. { =t 1Y D N.- I and hence there e x i s t s 1 m 1^ 1 1 1 m i 0 1 NXU-.UN+ no P-element i n G . Now we form a p a r t i t i o n { N ' N ' ...,N'} on N of length t < ( k) wit h the proper that f o r each N.' , there e x i s t s a p a r t i t i o n P.' on N.' 1 3 * 1 1 N.' such that no P/-element Is i n G 1 . Then by the same argument as In Theorem 5.11, G i s not a (k)-group N, which i s a c o n t r a d i c t i o n . Hence G x i s a (t)-group f o r a l l i . § 3. Imprimitive " t r a n s i t i v e (k)-groups of degree n , 3k < n < 4k . Theorem 5.15- Let G be of type \'\^ A L [ • Then G e (the subgroup c o n s i s t i n g o f ' a l l even permutations i n G ) i s an i m p r i m i t i v e t r a n s i t i v e minimal (k)-group of degree n = 4k . Proof: Let {N^,N2,N^,N4} be a complete block system e of G and l e t P = {X, ,X p, . . ,X, } be any p a r t i t i o n on N 85 I f |X.. n N . | _> 3 f o r s o m e i a n d j , t h e n a P -e e l e m e n t , w h i c h i s a 3 - c y c l e , c a n h e c h o s e n f r o m G I f | X . n N . I = IX. n N . I > 2 f o r s o m e i , 4 i c  x l J l X 2 J 2 - . 1 ^ o r j - , 4 J o > we c h o o s e x - , , x P € X . Cl N . a n d y - ^ Y g e x j _ n N j -. T h e n ( x 1 x 2 ) ( y 1 y 2 ) i s a p - e l e m e n t i n G e . F i n a l l y we c o n s i d e r t h e c a s e w h e n t h e r e e x i s t N . , J l N . a n d N . , w h e r e j' , j p a n d j" a r e a l l d i s t i n c t , w i t h J 2 J^ > ^ ] X . D N . I = IX. n N . | = | X . n N . I = 1 f o r a l l i . 1 i 3 ± 1 i J 2 1 i J 3 1 T h e n we c a n f i n d a n e v e n p e r m u t a t i o n g o f o r d e r 3 i n G e w i t h g ( N . ) = N . , g ( N . ) = N . . T h u s g J l J 2 32 J 3 i s a P - e l e m e n t a n d h e n c e G i s a ( k ) - g r o u p . B y u s i n g a n a r g u m e n t s i m i l a r t o t h e o n e u s e d i n t h e p r o o f o f T h e o r e m 3 - 5 , we m a y e a s i l y s h o w t h e Q m i n i m a l i t y o f G T h e o r e m 5 . 1 4 . L e t G b e a n i m p r i m i t i v e t r a n s i t i v e ( k ) - g r o u p o f d e g r e e n w i t h k _> 5 a n d 3 k + 1 <_ n <_ 4 k T h e n ( i ) G c o n t a i n s a s u b g r o u p o f t y p e A ^ + 2 » " \ + i * S k + i o r S k + 1 * S k + 1 * S k + 1 i f n < 4 k , a n d ( i i ) G i s o f t y p e (S]^\A^) e ! i f n = 4 k a n d G i s m i n i m a l . 86. P r o o f : L e t { N - ^ , N g , . . . , N g } b e a c o m p l e t e b l o c k s y s t e m o f G . s u c h t h a t t h e l e n g t h o f a b l o c k I s m i n i m a l . N . T h e n G i s p r i m i t i v e . F o r k > 5 , we h a v e n < k 2 - k + 2 = ( ( g ) + l ) - 2 a n d t h e r e f o r e b y C o r o l l o r y 3 . 3 , | ' N ^ | _> k f o r a X 1 i • S i n c e n <_ 4 k c V w e h a v e f u r t h e r t h a t s = 2 , 3 o r 4 . O n t h e o t h e r h a n d , b y N . N . T h e o r e m 5 .11 G 1 i s a ( k - 1 ) - g r o u p a n d h e n c e A l t ( N ± ) c G 1 f o r a l l i . We f i r s t c o n s i d e r t h e c a s e s < 3 • S u p p o s e t h a t A l t ( N ^ ) i s n o t c o n t a i n e d i n G f o r a l l i . L e t x 1 , x 2 , x ^ e N ^ , y e N g a n d A = N - { x - ^ x ^ x y } . T h e n G A i s a n i n t r a n s i t i v e ( k - 1 ) - g r o u p o f d e g r e e <_ n - 4 . L e t H b e a m i n i m a l ( k - 1 ) - g r o u p c o n t a i n e d i n G A . We t h e n h a v e t h e f o l l o w i n g : ( 1 ) I f H h a s 3 o r b i t s , t h e n H h a s a s u b g r o u p o f t y p e S^S^S^. b y L e m m a ^5.9 a n d T h e o r e m 5 . 7 . ( 2 ) I f H h a s 2 o r b i t s , t h e n H c o n t a i n s a s u b g r o u p o f t y p e S k * S k o r ( H k _ 1 , K , f 2 ) f o r s o m e K e 5 2 ^ H k - l ) ' b y T h e o r e m 5 - 3 a n d - 4 . 3 -( 3 ) . I f H h a s o n e o r b i t t h e n H c o n t a i n s a s u b g r o u p o f t y p e S R , A k + 1 o r H k _ 1 . 87-I n e a c h o f t h e a b o v e c a s e s , we c a n s e e t h a t G A c o n t a i n s a s u b g r o u p o f t y p e o r ^ . T h e n a s i m p l e c o m p u t a t i o n s h o w s t h a t G h a s a 3 - c y c l e . T h u s A l t ( N ^ ) c G f o r a l l i . T h i s c o n t r a d i c t s t h e a b o v e a s s u m p t i o n . T h e r e f o r e we h a v e A l t ( N ^ ) c G f o r a l l i . N o w , s i n c e A l t ( N i ) i s a ( k ) - g r o u p f o r | N 1 | _> k + 2 , we n e e d o n l y c o n s i d e r t h e c a s e s = 3 a n d n = 3k + 3 • I n t h i s c a s e | N ^ | f 0 (mod k ) a n d t h e n b y T h e o r e m 5 . 1 2 , N . N , G 1 i s a ( k ) - g r o u p . T h e r e f o r e S y m ( N ^ ) c G 1 a n d h e n c e b y L e m m a 5 . 6 G c o n t a i n s a s u b g r o u p o f t y p e S k , - ^ * S k ^ o r S k + l * S k + l * S k + l • T h i s P r o v e s ( i ) -N o w , we p r o v e ( i i ) . T h e o n l y c a s e t o c o n s i d e r i s s = 4 . T h e n e a c h N ^ c o n s i s t s o f k - e l e r n e n t s . L e t N± = { x 1 , . . . J x k } , N £ = { y 1 , . . . , y k } , N ^ = [z^ ... N 4 = ^ w l ' ' ' * , w k ^ a n ' ^ G - i& e . S y m ( N ) | g i s e v e n a n d g ( N ± ) = N ± , i = 1 , 2 , 3 , 4 } . We s h o w t h a t G * c G . I f n o t , we m a y a s s u m e t h e r e e x i s t \ x - ^ , x 2 e N ^ a n d y - ^ Y g e Ng s u c h t h a t ( x i x 2 ^ y l y 2 ^ ^ G ( n o t e t n a t i f G c o n t a i n s a t r a n s p o s i t i o n i n a n y b l o c k , t h e n b y t h e p r i m i t i v i t y o f N . G 1 , S y m ( N i ) c G f o r a l l i a n d h e n c e G i s n o t m i n i m a l ) . L e t A-^ = N - { x ^ , x 0 , y ^ , y 0 , z ^ } a n d A.., A 2 = N - { x 1 , x 2 , y 1 , y 0 , w 1 } . T h e n G a n d G ^ a r e i n t r a n s i t i v e ( k - 1 ) - g r o u p o f d e g r e e _< 4 ( k - 1 ) - 1 . S i n c e G h ^ s n o p r i m i t i v e s u b g r o u p o f d e g r e e _> k , i t i s n o t a ( k ) - g r o u p b y T h e o r e m 2 . 1 A l A 2 a n d T h e o r e m 2 . J . T h e r e f o r e e a c h o f G a n d G c o n t a i n s a s u b g r o u p o f t y p e S ^ . S i n c e z ^ e A ^ a n d A l A ? w ^ e Ag , we m u s t h a v e S y m ( N 4 ) c G a n d S y m ( N - ^ ) c G T h i s i m p l i e s t h a t S y m ( N ^ ) * S y m ( N / ( ) cr G . T h e n t h e t r a n s i t i v i t y o f G i m p l i e s t h a t S y m ( N ^ ) * S y m ( N g ) c G a n d h e n c e ( x - ^ X g ) ( y - ^ y g ) e G , w h i c h c o n t r a d i c t s t h e a s s u m p t i o n ( x - ^ X g ) ( y - ^ y g ) £ G . T h e r e f o r e G * c G . Now c o n s i d e r a p a r t i t i o n P = [ X - ^ X g , . . . , X ^ } o n N d e f i n e d b y X l = £ x l ' x 2 3 y l i Z l 5 W l ^ ' X 2 = { y 2 ' z 2 ^ W 2 ^ 5 X 3 = ( x 5 > y 5 > z 5 , w } X k = £ V W w k > L e t g b e a P - e l e m e n t i n G . W i t h o u t l o s s o f g e n e r a l i t y , we m a y a s s u m e t h a t g i s o f o r d e r 2 o r 3 . C a s e 1 . S u p p o s e t h a t g i s o f o r d e r 3 . T h e n g i s e v e n a n d G g = {g e S y m ( N ) | g i s e v e n , g ( N 1 ) = N 1 , g ( N . ^ ) = N . ^ , g ( N ^ ) = Nj. , { i 1 , i g , i 3 } = { 2 , 3 , 4 } } i s c o n t a i n e d i n G . We c o n s t r u c t a n o t h e r p a r t i t i o n 89. = { x 5 , y 5 , z 5 , w 5 } , . . . , Y k = { x k J y k , Z k , w k } . L e t h b e a P - ^ - e l e m e n t i n G . S u p p o s e h d o e s n o t * - 1 l e a v e N 1 i n v a r i a n t . T h e n < G ,g,h g h > c o n t a i n s a s u b g r o u p o f t y p e (S^\^A^)e . S i n c e G i s m i n i m a l a n d i \ \ I \ ) e i s a ( k ) - g r o u p , G m u s t b e o f t y p e ( S ^ A ^ ) 1 O n t h e o t h e r h a n d , s u p p o s e h l e a v e s i n v a r i a n t . T h e n G c o n t a i n s a s u b g r o u p o f t y p e Q k i f h i s o d d ( r e f e r t o t h e p r o o f o f T h e o r e m 'j>-5), a n d G c o n t a i n s a s u b g r o u p o f t y p e ^ k i s h i s e v e n . T h i s c o n t r a d i c t s t h e m i n i m a l i t y o f G . C a s e 2. S u p p o s e g i s o f o r d e r 2 . We m a y a s s u m e g ( N ^ ) = a n d g ( N ^ ) = . L e t h b e a P - e l e m e n t i n G d e f i n e d a s i n c a s e 1. -' I f h d o e s n o t l e a v e N~ D * -1 i n v a r i a n t , t h e n < G , g , h g h > c o n t a i n s a s u b g r o u p o f t y p e Q k o r ^ a c c o r d i n g a s g i s o d d o r e v e n , w h i c h c o n t r a d i c t s t h e m i n i m a l i t y o f G . O t h e r w i s e , N _ i s i n v a r i a n t u n d e r h a n d h e n c e • G c o n t a i n s a s u b g r o u p . o f t y p e ( H k J K 1 , f 1 ) * ( H k , K 2 , f 2 ) w h e r e K - ^ K g e 3 2 ( V a n d f 2 i s t h e o b v i o u s i s o m o r p h i s m . T h i s a l s o c o n t r a d i c t s t h e m i n i m a l i t y o f G . T h i s c o m p l e t e s t h e p r o o f . 91-C h a p t e r V I T h e ( k ) - c h a r a c t e r s o n m i n i m a l ( k ) - g r o u p s a n d  i t s a p p l i c a t i o n t o t h e t r i v i a l i t y o f a ( G , X ) - s p a c e . D e f i n i t i o n 6.1. L e t G b e a ( k ) - g r o u p . A s u b s e t K o f G i s s a i d t o b e a ( k ) . - s e t i f f o r a n y p a r t i t i o n P o n N ) A t h e r e e x i s t s a P - e l e m e n t i n , . . K . D e f i n i t i o n 6.2. A l i n e a r c h a r a c t e r x : G - F i s s a i d t o b e a ( k ) - c h a r a c t e r i f k e r x A K = (j) f o r s o m e ( k ) - s e t K i n G . ( B y k e r x we m e a n t h e k e r n a l o f t h e r e p r e s e n t -a t i o n o f G t h a t a f f o r d s t h e c h a r a c t e r x - ) L e t K ( G ) b e t h e s e t o f a l l m i n i m a l ( k ) - s e . t s i n G . T h e n i t i s c l e a r f r o m a b o v e d e f i n i t i o n s t h a t a ( G , x ) - s p a c e i s t r i v i a l i f a n d o n l y i f G i s a ( k ) - g r o u p a n d x i s a ( k ) - c h a r a c t e r o n G . i . e . k e r x fl K - (j) f o r . s o m e K e K ( G ) . I n t h i s c h a p t e r , we s h a l l d e t e r m i n e a l l t h e ( k ) - c h a r a c t e r s o n s o m e t y p e s o f m i n i m a l ( k ) - g r o u p s t h a t w e r e f o u n d i n t h e p r e v i o u s c h a p t e r s . We n e e d t h e f o l l o w i n g w e l l - k n o w n t h e o r e m o n c h a r a c t e r s . T h e o r e m 6 . ; . L e t x h e a l i n e a r c h a r a c t e r o n G , a n d l e t G ' x b e t h e d e r i v e d g r o u p o f G . T h e n k e r X 3 G ' a n d x i s i n d u c e d b y a l i n e a r c h a r a c t e r x o n G / G . 9 2 § 1 ( k ) - c h a r a c t e r s o n m i n i m a l ( k ) - g r o u p s o f t y p e S, - i a n d A, 0 k-i-1 k + 2 F o r k _> 3 > A k _ j g l i a s n o n o n - t r i v i a l l i n e a r c h a r a c t e r s . i F o r k = 2 , t h e d e r i v e d g r o u p o f A ^ i s = { ( 1 2 ) ( 3 4 ) , ( 1 3 ) ( 2 4 ) , ( 1 4 ) ( 2 3 ) , 1} a n d i t i s t h e o n l y n o n - t r i v i a l n o r m a l s u b g r o u p o f A ^ . S i n c e e v e r y ( k ) - s e t h a s n o n - e m p t y i n t e r s e c t i o n w i t h , t h e r e i s n o ( k ) - c h a r a c t e r o n A ^ . N e x t , we d e t e r m i n e t h e ( k ) - c h a r a c t e r s o n S, , . 3 K ' k + 1 We o b s e r v e t h a t t h e s e t K c o n s i s t i n g o f a l l t r a n s p o s i t i o n s o f i s t h e o n l y m i n i m a l k - s e t i n S ^ + ^ . T h e o n l y ( k ) - c h a r a c t e r X o n i s t h e f o l l o w i n g : 1 i f g i s e v e n X(g) = { 1 i f g i s o d d V g e S, k+I T h i s p r o v e s t h e f o l l o w i n g t h e o r e m . T h e o r e m 6 . 4 . L e t n < 2 k . A ( G , x ) - s p a c e U o v e r V i s t r i v i a l i f a n d o n l y i f G c o n t a i n s a s u b g r o u p H o f t y p e s k + 1 a n d t h e r e s t r i c t i o n o f x o n H i s t h e c h a r a c t e r d e f i n e d b y x ( g ) = 1 i f g i s e v e n a n d x ( g ) = - 1 i f g . i s o d d . -§ 2 . ( k ) - c h a r a c t e r s o n m i n i m a l ( k ) - g r o u p s o f t y p e H •k 93-L e t G = H k ( N 1 , N 2 ) w h e r e N-^ = ( x ^ x ^ . . . ; x k ) a n d N g = {y-^Vg, . . ,y^} . We f i r s t s h o w t h a t t h e d e r i v e d g r o u p G ' = S y m ( N 1 ) * S y m ( N g ) . I f g , h € G , t h e c o m m u t a t o r g - 1 h ~ 1 g h i s e v e n a n d g ~ 1 h ~ 1 g h ( N i ) = N i f o r i = 1,2 , a n d h e n c e G ' C S y m ( N ^ ) * S y m ( N g ) . O n t h e o t h e r h a n d , f o r a r b i t r a r y x . , x . e N , a n d y . , y . e N p t h e r e e x i s t s 11 12 1 J l J 2 a n e l e m e n t g i n G o f o r d e r 2 s u c h t h a t ( x . y . ) ( x . . y . ' ) i s a f a c t o r o f g . . T h e n 11 J l x2 J2 ( x x ) g _ 1 ( x x . ) g = ( x . x ) ( y y ) i s i n G ' a n d x l  L2 xl x2 xl x2 J l J l t h e r e f o r e G ' = S y m ( N 1 ) * S y m ( N 2 ) . S i n c e G / G ' = Z , ®  z 2 i s a g r o u p o f o r d e r l \ , we m a y w r i t e G / G ' = < g-^ > x < g g > w h e r e g ^ i s t h e c o s e t c o n s i s t i n g o f a l l o d d p e r m u t a t i o n s i n t e r c h a n g i n g t h e e l e m e n t s i n N ^ w i t h t h o s e i n N g a n d g g i s t h e c o s e t o f a l l o d d p e r m u t a t i o n s , g o f G w i t h g ( N ^ ) = N ^ . T h u s , t h e r e a r e t h r e e n o n - t r i v i a l l i n e a r c h a r a c t e r s v_]_ > X 2 A N D X-^ on. G / G 7 i n d u c e d b y t h e f o l l o w i n g : (1) x x (!]_) = 1 a ^ d x1Ci2^  = _ 1 > (2) * 2 (!-,_) = -1 a n d x 2 ( g 2 ) = -1 , (3) X^ dx) = -1 a ^ d X ^ ( g g ) = 1 . B y T h e o r e m 6.3 X;J_ , X 2 A N D X-^ i n d u c e c h a r a c t e r s X £ , x2 a n d . o n G r e s p e c t i v e l y . N o t e t h a t t h e s e t K c o n s i s t i n g o f a l l t r a n s -p o s i t i o n s o n N ^ a n d a l l t r a n s p o s i t i o n s o n N g , a n d a l l 94. p e r m u t a t i o n s o f o r d e r 2 ' w h i c h i n t e r c h a n g e t h e e l e m e n t s i n w i t h t h o s e i n N g , i s t h e o n l y m i n i m a l ( k ) - s e t i n G . T h e r e f o r e o n l y Gg-j_ g2 ° r G g l ^ a S e m ^ ^ i n t e r s e c t i o n w i t h K a c o r d i n g a s K i s o d d o r k i s e v e n . T h i s i m p l i e s t h a t x2 X s t l i e o n l y ( k ) - c h a r a c t e r / o n G i f k i s o d d a n d \ * i s t h e o n l y ( k ) - c h a r a c t e r o n G i f k i s e v e n . § 3 • ( k ) - c h a r a c t e r s o n m i n i m a l ( k ) - g r o u p s o f t y p e L e t G b e o f t y p e a n d l e t { N - ^ N g . N ^ } be- t h e c o m p l e t e b l o c k s y s t e m o f G . F o r x.. , x.. e N . x '"2 a n d y . , y . e N . , t h e r e e x i s t s a n e l e m e n t g e G s u c h J l J 2 3 t h a t g ( X . ) = y a n d g ( x ) = y . T h e n • xl J l  x2 J 2 ( x , x ) ( y y ) = ( x . x . ) g - 1 ( x x . ) g i s i n G ' a n d h e n c e J . 1 x2 J 1 J 2 J . 1 x2 j . 1 ± 2 c o n t a i n s t h e s e t o f a l l e v e n p e r m u t a t i o n s w i t h g ( N ^ ) = I^ L f o r i = 1 , 2 , 3 • M o r e o v e r , i t i s c l e a r t h a t G ' c o n t a i n s n o e l e m e n t g w i t h g ( N . ) = N . f o r s o m e i =j= j . H e n c e G / G ' i s o f - . o r d e r 6 . Now i f K i s a n y ( k ) - s e t t h e n K c o n t a i n s a l l t r a n s p o s i t i o n s o n N ^ f o r i = 1 , 2 , 3 , a n d a p e r m u t a t i o n g o f o r d e r 3 w h i c h i n t e r c h a n g e a l l t h e 95-e l e m e n t s i n t h e t h r e e s e t s , N 0 a n d . T h e r e f o r e K n G ' = (j) a n d K f l Ii 4 ty f o r a n y s u b g r o u p H o f G p r o p e r l y c o n t a i n i n g G ' . H e n c e t w o l i n e a r c h a r a c t e r s X o n G s u c h t h a t k e r x = G ' a r e k - c h a r a c t e r s o n G i f F i s a n a l g e b r a i c a l l y c l o s e d f i e l d o f c h a r a c t e r i s t i c 0 • §4. ( k ) - c h a r a c t e r s o n m i n i m a l ( k ) - g r o u p s o f t y p e Q, o r Q, k . L e t G b e o f t y p e Q a n d l e t B , , B 0 , B v , B „ , B , N-, , N 0 ' a n d N v b e d e f i n e d a s i n T h e o r e m 3 . 5 . 4 3 o 1 d j> We f i r s t s h o w t h a t B c G ' .• I n f a c t , f o r x-, , x 0 e N-, o — 1 d 1 a n d y^ , y 2 e t h e r e i s g e G s u c h t h a t g(x^) = y^ , • g(Xg) = y 2 a n d g | N ^ = 1 o r . g | N ^ = ( z-j_ z 2) f o r s o m e z r , z 2 e . T h e n ) ( z ^ ) g _ 1 ( x ^ ) ( z - ^ )g = ( x ] L x 2 ) ( y ] L y e G ' a n d t h e r e f o r e B c . G ' . F u r t h e r m o r e , o — J f o r e a c h e B 1 , h 2 e B 2 , h^h"1^  e G ' n B ^ a n d h e n c e B „ c B hrV^iho c G ' • T h u s G ' = B U B, . a n d Ur — O x d L d — O 4 G / G ' i s o f o r d e r 2 . N o w , l e t P = ' { X 1 , X 2 , . . . , X k } b e a p a r t i t i o n o n N w i t h X ^ D N ^ = {x-^,x23 , X 2 fl N 2 = {y-^,y23 a n d IX. D X . I < 1 f o r a l l ( i , j ) k { ( 1 , 1 ) , (2,2 ).} . T h e n c l e a r l y ( x i x 2 ^ y l y 2 ^ i s t h e o n - 1 - y P - e l e m e n t i n G . T h e r e f o r e a n y ( k ) - s e t K m u s t h a v e n o n - t r i v i a l i n t e r -s e c t i o n w i t h G ' a n d h e n c e ' t h e r e e x i s t s , n o ( k ) - c h a r a c t e r o n G . S i m i l a r l y , we c a n s h o w t h a t i f G i s o f t y p e , t h e n t h e r e e x i s t s n o ( k ) - c h a r a c t e r o n G . § 5 . ( k ) - c h a r a c t e r s o n i n t r a n s i t i v e m i n i m a l ( k ) - g r o u p s . L e m m a 6.5- L e t K b e a ( k ) - s e t i n G . T h e n K i s m i n i m a l i f a n d o n l y i f f o r e a c h g e K , t h e r e e x i s t s a p a r t i t i o n P o n N s u c h t h a t g i s t h e o n l y P - e l e m e n t i n K , M o r e o v e r , K i s t h e u n i q u e m i n i m a l ( k ) - s e t i n G i f a n d o n l y i f g i s t h e o n l y P - e l e m e n t i n G . P r o o f : I f f o r e a c h p a r t i t i o n P o n N , t h e r e i s a P - e l e m e n t g ' e K - {g} , t h e n K - {g} i s a ( k ) - s e t , w h i c h c o n t r a d i c t s t h e m i n i m a l i t y o f K . T h e r e f o r e f o r s o m e p a r t i t i o n P ^ o n g i s t h e o n l y P ^ - e l e m e n t i n K •. F u r t h e r m o r e , i f K i s t h e u n i q u e m i n i m a l ( k ) - s e t i n G , a n d i f t h e r e i s a P - ^ - e l e m e n t g ' i n G b u t n o t i n K , t h e n K ~ U { g ' } - {g} i s a ( k ) - s e t i n G n o t c o n t a i n -i n g K , w h i c h c o n t r a d i c t s t h e u n i q u e n e s s a n d m i n i m a l i t y o f 97-H e n c e g i s t h e o n l y P ^ - e l e m e n t i n G . C o n v e r s e l y , t h e p r o o f s a r e o b v i o u s . L e m m a 6.6. L e t G-^ a n d G g b e m i n i m a l ( k ) - g r o u p s o n a n d N g r e s p e c t i v e l y , a n d l e t K-^ a n d ' K g b e ( k ) - s e t s i n G ^ a n d G g r e s p e c t i v e l y . I f G = ( G 1 , L 1 , ( j ) 1 ) * ( G 2 , L 2 , < | ) 2 ) f o r s o m e L 1 e ^ ( G ^ a n d L g e 3 2 ( G g ) , t h e n K = ( K - ^ - L ^ ) ( K g - L g ) U ( K ^ I ^ ) U ( K g f l L g ) i s a ( k ) - s e t i n G . M o r e o v e r , i f b o t h a n d K g a r e m i n i m a l , t h e n K i s m i n i m a l , a n d I f b o t h K ^ a n d K g a r e u n i q u e m i n i m a l ( k ) - s e t s i n G ^ a n d G r e s p e c t i v e l y , t h e n K i s t h e u n i q u e m i n i m a l ( k ) - s e t ' i n G . P r o o f : F o r a n y p a r t i t i o n P = { X - ^ , X g , . . • o n N i u N2 > p i = C x i n N i ' - • ^ x k n N i ) a n d p2 = [ x 1 n N 2 , . . . , x k n N g } a r e p a r t i t i o n s o n N-^ a n d N g r e s p e c t i v e l y . T h e n t h e r e e x i s t P ^ a n d P g - e l e m e n t s g ^ a n d g 2 i n K ^ a n d K g r e s p e c t i v e l y . I f g^ e L ^ o r g g e L g , t h e n g ^ o r g g i s i n K . O t h e r w i s e , g ] _ g 2 £ 1^^2 a n d ^ e n c e g ^ g 2 e K . T h i s s h o w s t h a t K i s a ( k ) - s e t . N o w , we s h o w t h a t K i s m i n i m a l i f b o t h - . .K-^-- a n d K g a r e m i n i m a l . I f f a c t , f o r e a c h g ^ e K ^ , b y L e m m a 6.5 t h e r e e x i s t s a p a r t i t i o n P . = {X^,...,XX} o n N . s u c h t h a t g . i s t h e o n l y 1 J. K 1 1 P . - e l e m e n t i n K . , i = 1 , 2 S i n c e G . i s m i n i m a l , i l i t h e r e e x i s t s a p a r t i t i o n P ^ = [ Y ^ , . . . , Y ^ } o n I>L s u c h t h a t n o P ^ - e l e m e n t i s i n L i , i = 1 , 2 . N o w , i f 1 2 1 2 g l e K l n L l 5 t h e n p = { x i l j Y i J • • • , X k ^ Y k ^ i s a P a r t i t i o n o n U N 2 s u c h t h a t g ^ I s t h e o n l y P - e l e m e n t i n K . S i m i l a r l y , t h e r e e x i s t s a p a r t i t i o n P o n N-j U N 2 s u c h t h a t g 0 i s t h e o n l y P - e l e m e n t i n K i f g , , e K 0 f l Lr. . c d c d F u r t h e r m o r e , i f g-^gg e ( K i " L i ) ( K 2 ~ L o ) > t h e n g-^Sg i s t h e o n l y P - e l e m e n t i n K w h e r e P = [ X ^ u x f , . . 3X^\jXT.) . T h e r e f o r e K i s m i n i m a l . F i n a l l y , s u p p o s e i s u n i q u e m i n i m a l k - s e t I n G ^ f o r i = 1 , 2 . T h e n b y t h e s a m e a r g u m e n t a s a b o v e , we c a n s h o w t h a t f o r e a c h g e K t h e r e i s a p a r t i t i o n P o n N-^ U N,-, s u c h t h a t g i s t h e o n l y e l e m e n t i n G . T h i s p r o v e s t h a t K i s u n i q u e . L e t G ^ a n d G g b e t w o m i n i m a . l ( k ) - g r o u p s , a n d l e t G = ( G 1 , L 1 , ( ( ) 1 ) * ( G g , L g , ( } ) g ) , w h e r e L ± e ? 2 ( G i ) a n d ty :• Gj/11! 5 G i / / L i 5 1 = 1 > 2 ' L e t x : G - F b e a l i n e a r c h a r a c t e r s u c h t h a t k e r X 2 L g • D e f i n e X]_ '• G -» F b y p u t t i n g X-]_(g) = X ( g - ^ g g ) f o r a l l g e G . T h e n we h a v e . 9 9 -T h e o r e m 6 .J. x-j_  x s a l i n e a r c h a r a c t e r o n . M o r e o v e r , i f x i s a ( k ) - c h a r a c t e r o n G , t h e n x ^ i s a ( k ) - c h a r a c t e r o n G-^ . P r o o f : S u p p o s e  3 &±a2 £ G " T h e n ^-h'1 e L f= k e r X. a n d t h e r e f o r e x ( g x g 2 ) = X ( g ^ g ^ h ' ^ h g ) = X ^ h ^ g - j h g ) = X ( g 2 h ~ 1 ) X ( g 1 h 2 ) = x ( g - ] _ h 2 ) • H e n c e X-j^ i s w e l l - d e f i n e d . I t i s e a s y t o c h e c k t h a t X-, i s a l i n e a r c h a r a c t e r o n G1 . N o v ; , we s h o w t h a t x ^ i s a ( k ) - c h a r a c t e r w h e n e v e r x i s a ( k ) - c h a r a c t e r o n G L e t K b e a ( k ) - s e t i n G s u c h t h a t K D k e r X = ty , a n d l e t K-^ = {g-j_eG^ [ g - ^ g 0 e K f o r s o m e g 2 e G ] . T h e n K 1 i s a ( k ) - s e t i n G 1 a n d k e r X x fl K 1 = ty b e c a u s e k e r X IT K - ty"- H e n c e X i s a ( k ) - c h a r a c t e r o n G l • C o r o l l o r y 6 . 8 . T h e o n l y ( k ) - c h a r a c t e r o n S, , * S 1 k+1 "k+1" i s t h e c h a r a c t e r X w i t h k e r x = A , t 1 x A , , k+1 k+1 C o r o l l o r y 6.9- L e t G b e a ( k ) - g r o u p o f t y p e ( H k , L 1 , ( } ) ) * ( S 1 ( . + 1 , A k H ' 1 , ( } ) 2 ) . T h e n t h e r e i s a t m o s t o n e ( k ) - c h a r a c t e r o n G i n d u c e d b y t h a t o n H ^ . 1 0 0 . P r o o f : We o b s e r v e t h a t G ' D A , , . L e t y b e — k + 1 * a n y ( k ) - c h a r a c t e r o n G , t h e n x-^ , d e f i n e d a s i n T h e o r e m 6 . 7 , i s a ( k ) - c h a r a c t e r o n H , b y T h e o r e m 6.J. S i n c e we h a v e s h o w n i n S e c t i o n 2 o f t h i s c h a p t e r t h a t X-j_ i s u n i q u e , x i s u n i q u e a n d i s i n d u c e d b y x ^ • C o r o l l o r y 6 . 1 0 . L e t G-^ a n d G ^ b e t w o ( k ) - g r o u p s o f t y p e PL . L e t G = ( G , , L , ,d ) , ) * ( G 0 , L 0 , ( j ) 0 ) w h e r e L ^ e 3 ' 2 ( G i ) , i = 1 , 2 . T h e n t h e r e i s a t m o s t o n e ( k ) -c h a r a c t e r o n G . P r o o f : L e t G . = H, ( N , -. , N . r ) w h e r e i = 1 , 2 . T h e n t h e d e r i v e d g r o u p G 7 o f G c o n s i s t i n g o f a l l t h e e v e n p e r m u t a t i o n s g i n G . s u c h t h a t g ( N . .) = N . . , ±,j = 1 , 2 ( r e f e r t o s e c t i o n 2 o f t h i s c h a p t e r ) . T h e r e f o r e G / G 7 i s a n e l e m e n t a r y a b e l i a n g r o u p o f o r d e r 8 . H e n c e t h e v a l u e s o f a n y n o n - t r i v i a l ' l i n e a r c h a r a c t e r c o n s t i t u t e t h e s e t { 1 , - 1 } . L e t b e t h e u n i q u e m i n i m a l ( k ) - s e t c o n t a i n e d i n G ± i = 1 , 2 ( r e f e r t o s e c t i o n 2 o f t h i s c h a p t e r ) . T h e n b y L e m m a 6 . 6 , K = ( K ^ H L ^ U ( K g f l L g ) U ( K - L x ) ( K g - L 2 ) i s t h e u n i q u e m i n i m a l ( k ) - s e t i n G . N o w , l e t x h e a ( k ) - c h a r a c t e r o n G . T h e n x ( K ) = { -1 } , a n d s i n c e G 101 . i s g e n e r a t e d b y K , x i s u n i q u e l y d e t e r m i n e d . T h i s c o m p l e t e s t h e p r o o f . F i n a l l y , i t i s e a s y t o d e t e r m i n e t h e ( k ) - c h a r a c t e r s o n P S L(2,7) a n d D<- . T h e v a l u e s o f t h e ( k ) - c h a r a c t e r o n D,- c o n s t i t u t e , t h e s e t {1,-1} , a n d t h e r e e x i s t s n o ( k ) - c h a r a c t e r o n PSL(2,7,) b e c a u s e P S L(2,7) i s s i m p l e . We s u m m a r i z e a l l t h e r e s u l t s i n t h i s c h a p t e r b y t h e f o l l o w i n g t a b l e . 1 0 2 . T y p e s o f m i n i m a l ( k ) - g r o u p s J k + 1 A. k+2 H k ( i . e . S v ^ S p ) S k + l * S k + l k 2' N u m b e r o f 1 0 \ 1 [ k ) - c h a r a c t e r s S k\ A3 2 ( i f F i s a l g e b r a i c a l l y j c l o s e d o f c h a r a c t e r i s t i c 0)1 k ! 0 k 0 D 5 P S L(2,7) 0 X ( i i 1 0 ; B i b l i o g r a p h y W. B u r n s i d e , T h e o r y o f g r o u p s o f f i n i t e o r d e r . M a r v i n M a r c u s a n d H e n r y k M i n e , G e n e r a l i z e d  m a t r i x f u n c t i o n s , T r a n s . A m e r . M a t h . S o c . 1 1 6 ( 1 9 6 5 ) , 3 1 6 - 3 2 9 • K . S i n g h , O n t h e v a n i s h i n g o f a ( G , a ) p r o d u c t i n a ( G , g ) s p a c e , C a n . J . M a t h . , V o l . X X I I , N o . 2, 1 9 7 0 , p p . 3 6 3 - 3 7 1 R . W e s t w i c k , A n o t e o n s y m m e t r y c l a s s e s o f t e n s o r s , J o u r n a l o f A l g e b r a 1 5 ( 1 9 7 0 ) , 3 0 9 - 3 1 1 H . W i e l a n d t , F i n i t e p e r m u t a t i o n g r o u p s , A c a d e m i c P r e s s , New Y o r k , 1 9 6 4 . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080485/manifest

Comment

Related Items