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Involutions of the Mathieu groups Fraser, Richard Evan James 1966

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THE INVOLUTIONS OP THE MATHIEU GROUPS Richard Evan James Fraser B . S c , U n i v e r s i t y of B r i t i s h Columbia, i960 A THESIS SUBMITTED IN PARTIAL .FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department* of Mathematics We •accept t h i s t h e s i s as conforming t o the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1966. In p resen t i ng t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree that the 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 re fe rence and s tudy . I f u r t h e r agree that pe r -m iss ion f o r ex tens i ve copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s represen ta t i ves . , i t i s understood that copy ing o r p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department o f Mathematics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date February 8, 1966.. i i : ABSTRACT The f i v e Mathieu permutation groups M^, M i 2 * M22* M 2 3 and Mp^ are constructed and the i n v o l u t i o n s (elements of order two) of these groups are c l a s s i f i e d according t o the number of l e t t e r s they f i x . I t i s shown tha t i n M 1 2 ah i n v o l u t i o n f i x e s no l e t t e r s or f o u r l e t t e r s , w h i l e i n Mgij.- an i n v o l u t i o n f i x e s zero or eigh t l e t t e r s . I t i s a l s o shown t h a t i n each of the Mathieu groups, a l l the i r r e g u l a r i n v o l u t i o n s are conjugate and t h a t i n M 1 2 a l l the r e g u l a r i n v o l u t i o n s are conjugate. The orders of the c e n t r a l ! z e r s of the i n v o l u t i o n s are c a l c u l a t e d and i t i s shown t h a t no r e g u l a r i n v o l u t i o n l i e s i n the centre of a 2-Sylow subgroup. Most of the r e s u l t s are obtained by c a l c u l a t i n g d i r e c t l y the form a permutation must take i n order t o have a c e r t a i n property and then f i n d i n g one or a l l the permutations of t h i s form. , i i i TABLE OP CONTENTS I n t r o d u c t i o n 1 S e c t i o n 1 T r a n s i t i v e Groups and the T r a n s i t i v e Extension Theorem 3 S e c t i o n 2 C o n s t r u c t i o n of M 1 2 13 S e c t i o n 3 I n v o l u t i o n s of M^, M 1 2 2 4 S e c t i o n 4 C o n s t r u c t i o n of M 2 2, Mgy M 2^ 37 S e c t i o n 5 I n v o l u t i o n s of M 2 2, M^, Mg^ 47 B i b l i o g r a p h y 76 Appendix 77 i v ACKNOWLEDGEMENT The author wishes t o express h i s thanks t o Dr. Rimhak Ree who suggested the t o p i c , guided the research work and made many h e l p f u l suggestions during the p r e p a r a t i o n of the f i n a l d r a f t of t h i s t h e s i s . The f i n a n c i a l a s s i s t a n c e of the N a t i o n a l Research C o u n c i l of, Canada i s g r e a t f u l l y acknowledged. 1 INTRODUCTION I n t h i s t h e s i s we c o n s t r u c t , f o l l o w i n g W i t t [ 7 ] , the f i v e Mathieu permutation groups M^, M i 2 > M22* M 2 3 * a n < 3 M24* W e then c l a s s i f y the i n v o l u t i o n s (element o f o r d e r two) of these groups a c c o r d i n g to the number of l e t t e r s which they f i x . We a l s o c a l c u l a t e the number of i n v o l u t i o n s o f each type and study t h e i r e e n t r a l i z e r s . By c o n s t r u c t i o n , M 1 1 a M^ 2 a n d M 2 2 c ^23 c Mgij. so most o f the d i s c u s s i o n w i l l be about M 1 2 and M2^. The groups M 1 2 and M ^ were d i s c o v e r e d by Mathieu [5] I n 1861 [ 7 ] . They are 5 - f o l d t r a n s i t i v e on 12 and 24 l e t t e r s r e s p e c t i v e l y and a r e , except f o r symmetric and a l t e r n a t i n g groups, the o n l y known 5 - f o l d t r a n s i t i v e groups [ 6 ] . S i m i l a r l y , M ^ and M 0, are the o n l y known 4 - f o l d t r a n s i t i v e groups. The Mathieu groups are simple and a r e the o n l y known simple groups which do not l i e i n an i n f i n i t e f a m i l y o f simple groups [6.]. W i t t [7] g i v e s a r e c u r s i v e method f o r extending a doubly-t r a n s i t i v e group t o a k - f o l d t r a n s i t i v e group by a d j o i n i n g one new l e t t e r and one new permutation a t each stage. We prove t h i s t r a n s i t i v e e x t e n s i o n theorem i n d e t a i l i n S e c t i o n .1. I n s e c t i o n s 2 and 4 we use t h i s theorem t o c o n s t r u c t the Mathieu groups. The Mathieu groups are sometimes d e f i n e d as the groups o f permutations l e a v i n g i n v a r i a n t the s e t s of c e r t a i n S t e i n e r Systems [ 6 ] , [ 7 ] . I n p a r t i c u l a r M 1 2 and M ^ are the S t e i n e r Group of ( E T ( 5 , 6, 12) and < S ( 5 , 8 , 24) r e s p e c t i v e l y . T h i s p r o p e r t y i s used i n S e c t i o n 5 . 2 An i n v o l u t i o n i s s a i d t o "be r e g u l a r i f i t f i x e s no l e t t e r s . Otherwise i t i s i r r e g u l a r . I n Se c t i o n 3 we f o l l o w Wong's paper [8] and show that a l l the r e g u l a r i n v o l u t i o n s of M 1 2 are conjugate and a l l the i r r e g u l a r i n v o l u t i o n s are conjugate. Each i r r e g u l a r i n v o l u t i o n f i x e s f o u r l e t t e r s . We c a l c u l a t e the number of i n v o l u t i o n s of each type i n M^ 2 and the order of t h e i r c e n t r a l i z e r s . We a l s o show tha t each i r r e g u l a r i n v o l u t i o n l i e s i n the centre of some 2-Sylow subgroup of M 1 2 a-*1*3 t h a t no r e g u l a r i n v o l u t i o n has t h i s p r o p e r t y . F i n a l l y i n Se c t i o n 5 we d e r i v e analogous p r o p e r t i e s f o r the i n v o l u t i o n s of Mg^. We show tha t a l l the i r r e g u l a r i n v o l u -t i o n s of M2i|. are conjugate and t h a t each f i x e s e i g h t l e t t e r s . We c a l c u l a t e the number of i r r e g u l a r i n v o l u t i o n s and the order of t h e i r c e n t r a l i z e r s and show tha t each l i e s i n the centre of a 2-Sylow subgroup of M^. We are unable t o determine whether or not a l l the r e g u l a r i n v o l u t i o n s are conjugate. However we do c a l c u l a t e the number of r e g u l a r i n v o l u t i o n s i n and show that none l i e s i n the centre of a 2-Sylow subgroup of M2^. The r e s u l t s i n S e c t i o n 5 are, we b e l i e v e , o r i g i n a l and f o r tha t reason the most important p a r t of the paper. Most of the r e s u l t s are obtained by c a l c u l a t i n g d i r e c t l y what form a permutation must take i n order t o have a c e r t a i n property, and then f i n d i n g one or a l l the permutations of that form. 3 Sec t i o n 1. T r a n s i t i v e Groups and the T r a n s i t i v e  Extension Theorem. 1.1 D e f i n i t i o n . L e t G be a group of permutations on the l e t t e r s a,, .... a . L e t T c: {a,, .... a }. G i s k - f o l d t r a n s i t i v e l n 1* n on T i f and only i f f o r each p a i r of ordered subsets [b^, , b^} and ( c ^ , c^ ,} of T there e x i s t s an x e G such t h a t b^ — — > c^ f o r each j = 1, 2, k. I n t h i s s e c t i o n we w i l l prove f o r f u t u r e use f o u r general lemmas f o r t r a n s i t i v e groups. This I s followed by a proof of W i t t ' s t r a n s i t i v e extension Theorem. 1.2 Lemma. L e t G be a group of permutations, on the l e t t e r s of a set T. I f there e x i s t s an ordered subset { r ^ , . r f e } c T such t h a t f o r each ordered subset { t ^ , t^} c T there e x i s t s x.e G such t h a t t ^ — - — > r ^ ? i = 1, , k then G i s k - f o l d t r a n s i t i v e on T. Proof. L e t [t^, t k ] and {u^, u^} be any two ordered subsets of k l e t t e r s of T. There e x i s t x, y € G such t h a t t ^ — x > r ^ and u ^ — ^ — * " r i ^ o r -^= -^> 2> k ' Hence t ^ ^ — > . u^ , i = l , k and thus G i s k - f o l d t r a n s i t i v e on T. 1.3 Lemma. L e t G be a group of permutations which i s k - f o l d t r a n s i t i v e on the l e t t e r s of a set T. L e t H = {x : x e G and x f i x e s r ^ , r 2 , r f c e T]. Then H i s a group and i f H i s £-fold t r a n s i t i v e on the l e t t e r s of T - [r^, , r f c} then G i s (k+-l)-fold t r a n s i t i v e on T. 4. P r o o f . L e t x, y e H. Then x, y and y " 1 f i x r ^ , , - r ^ . Th e r e f o r e x y " 1 f i x e s r ^ , ..., r k and thus x y " 1 e H. T h e r e f o r e H i s a group. Assume H i s . i - t o l d t r a n s i t i v e on T - [r^, , r k } . L e t {t^,...,\>\+i> • • • <= T be any ordered s e t o f k+t l e t t e r s . There e x i s t s g e G such t h a t t ^ — - — ? - r ^ i = l , k s i n c e G i s k - f o l d t r a n s i t i v e on T. L e t u^ be such t h a t t ± ^ - ^ u i i ^ + l , ..., k+l. Por any r k + 1 , ..., r k + ^ e T -{r^, ..., r f c} t h e r e e x i s t s an h € H such t h a t u^ ^ r ± i=k+l, ..., k+l (and h f i x e s r ^ , r f e ) . Hence t i ^ ^ r i — — - ^ r i i = l , ..., k t± u ± r h > r± i=k+l, ..., k+l and g h 6 G. T h e r e f o r e G i s (k+-e.)-fold t r a n s i t i v e on T. 1.4 Lemma. I f G i s k - f o l d t r a n s i t i v e on a set T of n l e t t e r s and i f .H = fx ': x e G, x f i x e s some [ r ^ , ..., r k J c T] then |G| = n ( n - l ) ... (n-k+1) |H|. P r o o f . There are u = n ( n - l ) ... (n-k+1) d i s t i n c t ordered subsets o f T each w i t h k l e t t e r s . L e t A,, — , A be these ordered s e t s w i t h A^ = {r^, ..., r f c } . F o r each x i 1=1, | i t h e r e e x i s t s x^ € G such t h a t A^ ^A^ because G i s k - f o l d t r a n s i t i v e . F o r any g € G, g maps A^ ^ onto some A j . Hence SXj -" 1" maps A^ onto i t s e l f . Thus g X j " " 1 e H and so g e H x ^ . A l s o , Hx i n Hx^ = 0 i f i ^ j \ hx. hx. because A, ^»-A. and A, .^ A j f o r each h e H. Therefore G = Hx 1 + Hx 2 + ... + Hx^ (coset decomposition) and (G| = u |H| = n ( n - l ) ... (n-k+1) |H|. 1.5 Lemma• L e t G 2 be a doubly t r a n s i t i v e group on T 2 = kl» pr> q l > q 2 ^ * L e t ^ = {x j x 6 G 2 and x f i x e s q 2) and H = {x ; x e G 2 and x f i x e s q.^  and q 2 Then i ) , a n d H are groups. i i ) i i i ) i v ) v) G 1 i s ( s i n g l y ) t r a n s i t i v e on T 1 = T 2 - [q 23. There e x i s t s S 2eG such t h a t S 2 = P 2 ( q 1 , q 2 ) where P 2 i s a permutation f i x i n g q 1 and q, G 2 = G 1 + G ^ S ^ S2 e H. Proof, i ) I f x, y e G 1 then x and Therefore y - 1 f i x e s q 0 and xy y n x q 2 . -1 and x y - 1 s. G ^. Hence G-^  f i x f i x e s i s a group. S i m i l a r l y H i s a group, i i ) L e t u, v € 1^ c T 2, u v. There e x i s t s x e G 2 such t h a t q 2-u-x ->q0> —>• v since G 2 i s doubly t r a n s i t i v e on T 2 < But q, x l 2—->>q 2 i m p l i e s x e ( s i n g l y ) t r a n s i t i v e on T^. Thus G 1 i s i i i ) There e x i s t s x e G g such t h a t q 1—^>>q 2, q 2 X >q^ sin c e G 2 i s doubly t r a n s i t i v e . Thus x permutes the p^'s amongst themselves, and so x = P 2 ( q 1 ^ q 2 ) where P 2 i s a permutation f i x i n g q^ ^ and q 2 ( i . e . permuting only the p ^ ' s ) . Define S 2 = x. L e t x € Gg. I f q g X> q 2 then x e G^ I f q g xy.m^q 2 . then u e.Tj-' L e t v e such t h a t v — x - * - q 2 > There e x i s t y., z c such that u y ^ q ^ q ^ L — i ^ v since G^ i s . t r a n s i t i v e on . Thus q x Z > v — ^ q g - ^ ^ q g and q 2 _ ^ q 2 _ x ^ u ^ ^ q 1 Hence q 1 < Z X y > q 2 . Thus zxy = QCq^ q^g) ^ o r some Q which f i x e s q 1 3 q 2 . So zxy S" 1 = Q(q1,q2) (q^qg) ^ P g 1 - Q Pg" 1 e G 2 Therefore Q P g " 1 = g e G x since each f i x e s q 2 But zxy Sg" 1 - g i m p l i e s x =.'(z-"1g)Sg y" 1 e G^^G Therefore G g . c G ^ t G^gG-^ But G 1 c G 2 and S 2 e. G2- Therefore G ^ S g ^ c G Therefore G 2 - G 1 + G^-pG^ . S 2 2 = P 2(q 1 ,q 2) PgCq^q'g) - P 2 2 ( V q 2 ) 2 = P 2 2 € G 2 2 2 Therefore S 2 e H sin c e P g f i x e s q-j^  and q 2 . 7. 1 . 6 Theorem. ( T r a n s i t i v e Extension Theorem) Le t Go be a group which i s doubly t r a n s i t i v e on T 2 = { p 1 } P r.q 1>q 2 3 and l e t S 2 = P 2 ( q 1 , q 2 ) e G 2* P 2 f i x e s q 1 , q 2 • Let G-L = {x ; xsG 2, x f i x e s q 2 3 H = {x : xeG 2, x f i x e s q 1 and q 2 } . Por each k = J>, . t a d j o i n a new l e t t e r q k to form \ - T k - 1 U {q k} and a permutation Sfc = P k ^ k - l ' q k) where P f e f i x e s q i , . . . , q k . L e t G k = G ^ + G ^ ^ G ^ . I f i ) S k c H k=3i • • • » t i i ) ( S k - l S k ) 3 e H k=3,.-.,t i i i ) ( S ^ S k ) 2 e H ^ < l < k-2, k = 4,...,t S kG^S k = G-^  k = 3, . . ., t . Then i ) G K i s a group and i s k - f o l d t r a n s i t i v e on T k, k=3,•• i i ) HS k — ^ kH k=2y . . . , t i i i ) G j S k = S k G j 1 < J < k~ 2> 3 < K < t i v ) G k = {x : x€G t, x f i x e s k=l, . •», t This theorem w i l l be proved i n s e v e r a l p a r t s . I n each p a r t , we s t a t e the p r o p o s i t i o n and then give the proof. (1) = ^k^ k=2, ..., t . Proof, i f k=2, l e t heH. SghS* 1 = P 2 ( q 1 3 q 2 ) h ( q 1 , q 2 ) ~ 1 P 2 ~ 1 = P 2 h P' 1 e G 2 S ^ S " 1 = € H since P 0 and h both f i x q n,q«. 8. Hence S 2 H S" 1 c H. Thus Sn H S" 1 = H since both have the same number of elements, and so S 2 H = HSg. For K 2 5j Sfc G 1 S k = G1 by hypothesis. Hence S kH e Sifl1 = S k ( S k G 1 S k ) c H G.^ since S k 2 e H = G^Sjj since G^ i s a group. Therefore, f o r each heH there i s a geG^ such that q x *- q 1 9 - q-j^  g since S k f i x e s q 1,...,q k_ 2. q1—^-^u k ^ v But v = since S kh = g ^ . Hence u = q^ since S k f i x e s q^. Therefore g f i x e s q^ (and .geG^)- so geH. Thus SkH c H S k , and so S kH = HS k since |SkH| = |HSfc|. (2) I f G^ i s a group then ; S, G. = G..S, i f f S.G, S, = G. T i i k T i k i Proof. S k 2 e H. Thus ^ = h ^ - 1 = S^"^ f o r some heH. S k G i = G i S k l f f ' V ^ i ~ G i h S k _ 1 = G i S k _ 1 s i n c e G i i s a S r° uP i f f S kG,S k . C,, (3) G k i s a group and S kG^ = G^S^, k=5,...,t 1 < 1 1 k~2-Proof. G 2 i s a group and S f c G i S k = G i ^ hypothesis (k>3) Therefore S kG 1 = G.^ (by p a r t (2)) f o r each k>>. Assume tha t f o r some k>3 and some n, 2<n<k-l, Gg, .. • ^ j c„ 1 are groups and G^S^ = S^G^ f o r j= 3 , ,k and 1=1, . . . , 3 - 2 and G^^S^ = S f c + 1G i f o r 1=1,...,n-l G k G k = ( Gk-1 + G k - l S k G k - l ^ G k - l + G k - i s A . l ) ( d e f i n i t i o n of G k) = ( G k - l G k - l + G k - l G k - l S k G k - l + G k - l S k G k - l G k - l ) + G k - l S k G k - l G k - l S k G k - l = G k + G k - l S k G k - l S k G k ~ l ( s i n c e G k - 1 i s a § r o u P ) = G k + G k - l S k ( G k - 2 + G k - 2 S k - l G k - 2 ) S k G k - l = G k + G k - l S k G k - 2 S k G k - l + G k - l s k G k - 2 S k - l G k - 2 S k G k - l 2 = ( Gk + \ - l \ - 2 S k G k - l ) + G k - l G k - 2 S k S k - l S k G k - 2 G k - l ( s i n c e S kG k_ 2 ** G k - 2 S k fey t h e i n d u c ' t i o n assumption) = ( G k + G k - 1) + G k . 1 S k S k _ 1 S k G k _ 1 ( s i n c e G k ^ i s a group by t h e ' i n d u c t i o n assumption) But ( s k _ i s k ) ^ € H a n d ( S ^ ) 2 e H f o r e a c n s o S k S k - l S k G S k l l H c ( S k - l H ) ( S k H ) ( S k - l H ) H = S ^ S ^ H (si n c e S ±H = by Pa r t 1) . T h u s G k G k c G k + ^ ( ^ A . ^ ) ^ = G k + G k_ 1S kG k_ 1 = G k so G k i s a group ( S n S k + 1 ) 2 s H since 2 _< n ( k + l ) - 2 and ( S ± ) 2 € H f o r a l l i . 10. L e t ( s n W 2 - h i > s n 2 - h 2 ; s k + 2 - h 5 ; h i e H S k + l S n S k * = Sn" 1 h l = S n h i l h l = S n h 4 ? hk = h l \ 6 H ' S k + l ^ k + l = S k + l ^ G r i - l + G n - l S n G n - l ) S k + l = S k + l G n - l S k + l + S k + l G n - l S n G n - l S k + l 2 = Sk+1 Gn-1 + G n - l S k + l S n S k + l G n - l ( S k + l G n - l = G n - l S k + l b y i n ( 3 u c t i - o n assumption) = h,G w -i . + G -i S h,,G . , 3 n - l n - l n 4 n - l — ' Gi„ n + G„ i S_G_ n n - l n - l n n - l = G n (and G n i s a group). Therefore S f e + 1G n = G n S k + 1 (by P a r t ( 2 ) ) . (4) G k = {x : xeG t, x f i x e s • f o r e a c n k =X j • • • ^ \J ~*1 * P roof. For each k £ t - 1 , G k • e {x : xeG t, x f i x e s q k + 1,...,q t 3 because elements of G k are permutations on T k and can a l s o be considered t o be permutations on T^ = T k U io^^t • • • * ( l t ^ which f i x q k + 1» •••>q^• L e t xeG^ such t h a t x f i x e s q^. x € G t_ 1 or x e G t_ 1S tG t_ 1 since G t = 0 T - 1 + G ^ S ^ ^ . I f x = g 1 S t g 2 , g x , g 2 e G t_ 1 then q t g l > q t _ S - t ^ . q ^ g 2 >.u^  contrary to the choice of x f i x i n g q^ .. Therefore x £ G t _ i s t G t -so x c G t ^ and Gt_^ = {x : xeG^, x f i x e s q^.}. 11. Assume tha t f o r some n<t-2, G k = {xjxcG^., x f i x e s q k + 1 , . . .,qt} f o r each k = n + l , . . . , t - l . L e t xeG^ such t h a t x f i x e s q n + 1,...,q^. x f i x e s q n +]_ s o x £ Gn+1 fey t i i e i n d u c' t- t- o n assumption. I f x g l S n + 1 g 2 ,. g l,g 2.8 G n then g l S n . T So i %+i— * V — ' • " K + i contrary t o the choice of x f i x i n g q ^ ^ * Therefore x k G S ,,G„ sO x e G„ since G„,, = G + G S - G _ . Hence r n n+1 n n n+1 n n n+1 n G n = {x : x€G t, x f i x e s q n + 1>.•.,q t3• (5) G k i s k - f o l d t r a n s i t i v e on T k ?k=3,...>t. Proof. G 2 i s doubly t r a n s i t i v e on T 2 by hypothesis. Assume that f o r some k>3, Gj i s j - f o l d t r a n s i t i v e on T^ f o r each j = 2 , . . . , k - l . L e t u s Tfc. ' I f u = q k then u—^->u=q k and l c G ^ . I f u ^ q R then u e But G f c - 1 i s ( k - l ) - f o l d t r a n s i t i v e on T k_ 1 (by i n d u c t i o n assumption) and thus ( s i n g l y ) t r a n s i t i v e on T^^. Hence there e x i s t s 3 r eG k_ 1 such t h a t u y > q k M l . L e t x = y S k e G^. Then u - J - ^ q ^ , ^ > q k . Therefore f o r any u s T k there e x i s t s x € G k such t h a t u— X-**q k' Hence G k i s t r a n s i t i v e on T k by Lemma 1.2. But G k_ 1 i s ( k - 1 ) - f o l d t r a n s i t i v e on T k - 1 by the i n d u c t i o n assumption. Therefore, by Lemma 1.3, G k i s k - f o l d t r a n s i t i v e on T k. 12. 1.7 Remark. .Witt-[7]- a l s o proves the f o l l o w i n g theorem: I f G t i s t - f o l d t r a n s i t i v e on = {p-^.. .,vrt\» .. '>qT3 and G f c = {x j xeG^, x f i x e s q k + 1,...,q^} k = 1, . t - 1 • and H = {x : xeG^, x f i x e s q^, q^} Then i ) Gfc i s k - f o l d t r a n s i t i v e on T k {P]_, ••. .,pr>q-j_, •. .,q k) i i ) For each k=2,...,t there e x i s t s s k e G k such th a t s k ' s a ' p k ( q k - l , q k ^ P k f i x e s i i i ) S k 2€H ;k=2,. „. ,.t, (Sj^^S^) 5eH^ k=3, ..., t , ( S j S k ) 2 e H , • k-4, ,...,t J-2,.,.,k-2 i v ) G k = G k_ 1 + G k_^S kG k_ 1 k=2,...,t v) SfcH = ES^P k-2 3...,t v i ) S RGj = G^S k, k=3,...,t j=l, . . . , k - 2 . Thus any t - f o l d t r a n s i t i v e group, t>3, i s obtainable from a doubly t r a n s i t i v e group by means of the t r a n s t i v e extension theorem. 13 . S e c t i o n 2. C o n s t r u c t i o n of M^, M i 2 * 2.1 D e f i n i t i o n , i ) Q = PSL ( 2 , 9 ) «• SL ( 2 , 9)/Z where Z=centre of .,SL ( 2 , 9 ) . i i ) L e t a be a p r i m i t i v e element of GF(9) 8 2 (Hence a = 1, a+a = 1) f o r y e GF ( 9 ) , yA> Pk k = a = k & ) • < f ' ~ ) % = 1 = -(J) 1 = 0 = (?) = 0 q 3 — 00 — <0> lo» J q 4 = v. ft GP(9). q 5 = w ^  GF(9), w^v. Note: 7* 1* • • •*"5 can be considered to be the ten p o i n t s of a p r o j e c t i v e l i n e over GF ( 9 ) . i i i ) T k .= ( P l, .. .,p 7,q 1, .. .,qk} k^ L,-......5 T o = frl* ' " > ^ ' i v ) The f o l l o w i n g are permutations on T,_. x - ax j f i x e s <=,v,w - 1-x ? f i x e s oo,v,w (xeGF(9)) x -» X » p f i x e s v,w 2 . 3 x - a x+ax } «><£r->v f i x e s w. u 5 v) K D E - , 3 . v<->w f i x e s = <Q,V> = group generated by Q and y. - [a to s= = . {a : a = a 0 a a b 0 a -1 -1 a / 0 , aeGF(9)} a^O, a,beGF(9)}• 14. 2.2 Remark. In the following propositions i t w i l l he shown that E + Ey » G 2 i s doubly t r a n s i t i v e on T 2, the S^'s f i t the conditions'of .the.transitive extension theorem, and K = (Jg+G-gS^ Gr, — G^. Then and M^2 . a r e defined to he the t r a n s i t i v e extensions G^ and G^ (and are thus 4- and 5-fold t r a n s i t i v e on T]j_ and T^ respectively.) . These r e s u l t s are summarized i n theorem 2.1J. 2.3 Proposition. Z = fT (J °) , ("J. and thus (c a' = C c la) i n «• Proof. I f oeZ ( i . e . a commutes with each X G Q ) then a = (Q °) f o r some aeGF(9) such that a 2 = 1. Therefore a=l or -1 and Z i s as stated. Therefore i n the f a c t o r group Q = SL(2,9)/Z we have /a t>\ /-a -b\ ^c d' ^-c -d ; ' , 4 - 1 3 2 2.4 Proposition, y =1 and y : x --ax^ and y e Q« Proof. For x € GF(9), Y 3 Y / 3\ 3 4 v 3 n ' / 3 \ 3 4 = -a x = x. 4 Thus x— i— > x f o r each xsGF(9) and y f i x e s «,v,w. So 4 n , ^1 3 3 Y = 1 and y = y i x—>-axy. . L e t u = f°2 °0\ 6 Q \0 a / /xl |j ^  /a 2x | = ("M - « (~x) f o r a n y x ' y 6 G F ( 9 ) \y/ \a"" 23r/ \ 7 I , \ 7/ n o t both zero . Thus u : x—>-x, « ) * _ > o o . Therefore u = y 2 e Q. 2.5 P r o p o s i t i o n . K = Q + Qy. Proof. L e t X = ^ Jjj) e Q. det X - ad - be = 1 /x\ Y"" 1.^ j-gx^j \A~ax\ + b y 3 j y la[-ax\ + ^ y' ^ " 3 ' U x 3 c + dy 3/ \ [-ax 3c + y 5 d ] 5 , |a[-a. xarr + ybr 3 3 3 -a xc + yd-^  ' a \ + ab5y\ = / a 5 ab-5 \/x Wa7c3r + d5y/ \ a 7 c 5 d3 L e t p = a , a ^ I and so det p = a^d 3 - a^b^c 3 = (ad - be) p = 1 -1 • • -1 -1 Therefore y Xy = p € Q so y Q y c Q . Hence y QY •= Q since | Y " I L Q Y ! = Y ^ = 1 and y 2 € Q . L e t o e K. X^Y X 2 Y • • • • Y X N f o r some n where x^ € Q since a K =-<Q,Y> Qa = Q X ^ Y ^ 2 y * * * - Q Y X 2 Y • • • Y X N - Y Q X 2 Y ••• • Y X N - Y Q Y X 3 — yxn 16. - Qx5 . . . Y . ^ Q i f n i s odd Qy i f n i s even. Therefore K = Q, + Qy (coset decomposition) and Q i s a normal subgroup of K. 2.6 D e f i n i t i o n . G 2=* (a : CT € K, a f i x e s «> «= q^} G^ = {a t a e K, CT f i x e s » =* q^ and G = q 2] H =« {CT : CT e K, cr f i x e s <*>,031 ( i . e . q-j,q 2,q 1)}. 2.7 P r o p o s i t i o n . G 2 = E + Ey, Q1 = D + Dy, H = Cl} ; . sro*. L e t (-j) « , -•-(5)-*(5),o-(;)^.(S}-CT f i x e s oo i f f c » 0 ; a j= 0 i f f CT € E CT f i x e s » and 0 i f f c = 0 and b = 0 ; a ^  0 I f f CT € D. cry f i x e s co i f f a f i x e s » (since Y f i x e s °») i f f CT € E. CTY f i x e s oo and 0 i f f CT f i x e s » arid 0 ( s i n c e y f i x e s wand 0) i f f CT € D. But G 2 c K Q + Qy. Therefore G 2 = E + Ey and G 1 = D+Dy. By d e f i n i t i o n , H e G-^  = D + Dy. 17. Hence a e D and a f i x e s 1 i f f a = 1 i f f a = 1 o r -1 i f f a = 1 ay € Dy and ay f i x e s 1 i f f a 2- Y — a a - ^ = 1 2 i f f a = a which i s i m p o s s i b l e s i n c e a i s a p r i m i t i v e element o f GF(9). T h e r e f o r e H = {1}. 2 . 8 P r o p o s i t i o n . Q 2 i s a group and i s doubly t r a n s i t i v e on T 2 . P r o o f . I f of X € G 2 c K, then a and X f i x » and thus X" 1 and a X _ 1 f i x «. Hence aX" 1 € G 2 so G 2 i s a group. L e t x , y c T 2 = {a,.. . , a 7,l,0} m GF(9) x ^ y . x - y |» 0 so x - y =» f o r some j . Case I . I f x-y = a 2 k f o r some k l e t a= | a ' * " a e E 18. Hence x — ^ - s - O , y g Y > 1 • T h e r e f o r e f o r any x,y e T g ; X f'y . t h e r e e x i s t s X e G 2 = E + Ey such t h a t x X - s * 0 , y X > 1 . Hence G 2 i s doubly t r a n s i t i v e on T 2 by Lemma 1.2. 2.9 P r o p o s i t i o n . K = G 2 + GgS^Gg. P r o o f . L e t |j 0 a \ - a " 2 0 / € Q. -a 2 x .V/ = (?) -1 Hence : x—>x , 0-^->», so n = S, . L e t a 3 a b .6 Q. I f c = 0 then CT e E. I f c + 0 , l e t x 1 0 x-^s^x2 c d \6 c -1 \c d/ e E, 2 -2 -r a a ac € E. a 2 -2 -1^ a a ac a 0 a -2 -a c d ,0 c -1 rac 1 -1 0 c d a c _ 1 ( a d - l ) 0 c -1 a b\ , ; , s i n c e c~ (ad -1) = c~ (ad-ad+bc) ic d b s i n c e CT e Q. T h e r e f o r e Q c E + ES,E 3 SO K a Q + Qy c E + ES^E + Ey + ES^Ey C (E + E Y ) + (E+Ey) S 5 ( E + E Y ) G 2 + G 2 S 3 G 2 * 19. But S j € Q c,K arid Gg c L Hence Gg + GgS-^Gg c K, and K » G 2 + GgS^Gg. 2.10 P r o p o s i t i o n . S k 2 1 e H k = 2, 5 ( S k - l S k ) 3 * 1 k - 3, 4, 5 ( S ^ ) 2 = 1 n » 2, . . k - 2 k = 4, 5-P r o o f . S 2 = (a,a 2)(a 3 , c x 6)(a 5,a 7)(l,0)(a 2 ,)(»)(v)(w) m (a,a 7) ( a 2 , a 6 ) ( a 3 , a 5 ) (0,-) (1) ( a 4 ) (v) ( w ) 5 4 - (a, a 2 ) (a3,a7) ( a 5 , a 6 ) («, v) (1) ( a 4 ) (0) (w) 5 5 - (a,a 3 ) ( a 2 , a 6 ) ( a 5 , a 7 ) ( v , w ) (0) (1) ( a 4 ) (-) Thus 'Sg 2 * S^ 2 • • S j ^ 2 » S,~2 =* 1 s i n c e each i s a Product o f t r a n s p o s i t i o n s . ( 5 2 . 5 3 ) (a.,a 6,a 5) ( a 2 , a 7 , a 5 ) ( a 4 ) (l,-,0)'(v) (w) (S. 5,S 4) - ( a , a 5 , a 6 ) ( a 2 , o 5 , a 7 ) ( a 4 ) (1)' (0,v,-) (w) .•(S 4,iS 5) - ( a , a 6 , a 7 ) ( a 2 a 3 , a 5 ) ( a 4 ) (1) (0) (-,w,v) Thus ( S 2 , S ^ ) 3 = ( S y S ^ ) ' = ( S ^ j S ^ ) 3 = 1 s i n c e each i s a product o f t r i p l e s . ( 5 2 . 5 4 ) - (a)(a 2)(a 5,a 5)(a 4)(a 6,a 7)(l,0)(»,v)(w) ( S 2 > S 5 ) - (a,a 6)(a 2,a 5)(a 4)(a 5)(a 7)(l,0)(«)(v,w) ( 5 5 . 5 5 ) - ( d i a 5 ) ( a 2 ) ( o 5 , o 7 ) ( a 4 ) ( o 6 ) ( l ) ( 0 - , - ) . ( v , w ) 2 2 P Thus (S 2,S^) = (S 2,S^) = (SyS^) = 1 s i n c e each i s a product of t r a n s p o s i t i o n s . 20. 2 . 1 1 P r o p o s i t i o n . S^S^. = G 1 k 3 , 4, 5. P r o o f . L e t a =* | a ° M > 1j s D c G][. a and y both f i x (q 2> • • " J * ^ ™ t o , v , v r } while f o r each H , f i x e s two of them and interchanges the other two. Hence S k a S k a n d S k a y S k f l x °"*"» v» w' F o r x 6 T i * x \ a f a x a^x. L e t X =* f a ' 0• ,| e D. x — ^ * a ^ x so S,aS, =* X G D. 0 a"-5/ ^ ^ Thus S..DS, c D so S,DS-, = D since IS,DSJ =* |D|. S 3 C T 2 - 1 v 6 -3 3 . - 1 -6 3 6 2 3 , 2 6 x 3 x ^-?-a x — — "L-xxa x ——=^>a a x = o>a a xy =» a ( a a x) . T 4 . / o a 3 0 \ „ p 2 6 y / 2 6 N 3 L e t p = ( \ £ D. x—^->-a a x — ^ 0 . ( 0 a . x ) ^ so SjOySj •=» PY c DY. Hence S^ByS^ c By so S^DyS^ =* By, since | S ^ PY S ^ | » JDY| . Therefore S ^ S ^ =» S^DS^ + S^DyS^ .» D + Dy =< G l ' ^4 2 3 CT 2 2 3 x >a x + ax- a (a x+ax^) ^4 If ? 3 2 3 7 6 3 4 6 —>a a x + a a x + a a x ^ + a a x -a 2(l+a^)x + a 5 a 2 ( l - a 4 ) x 5 P Pk a x i f a = a . a ( a 2 a 6 x ) 5 i f a - a 2 k + 1 Co i f a - a 2 k Therefore S L F CT S 4 = <> ^ PY i f a =s a So Si, a Su 6 D + By. 21. S 4 ° 2 2 2 3 Y 7 6 3 4 6 x > a a x + a a x - — ± * - a a x^ + a a x - s 4 w . 6 3 6 6 '22.18'. 13 18 3 r—5*aa x^ + a a x + ct a- x + a a x-^  .=» a^a 2x(l+a 4) + a a 2 x ^ ( a 4 - l ) a a x i f "a"» a a ( a 6 x ) 5 i f a - a 2 k + 1 - f x i f a = a 2 k where T - (aa ° , -,) sD T h e r e f o r e S^, CTY S 4 - 1 . \ Q ^ V 1 / (_XY i f a = c* + 1 so cry S|j_ e D + By. Th e r e f o r e S ^ S ^ = S^DS^ + S^DyS^ c D.+ Dy * G r so S 4 G 1 S ^ - - G 1 s i n c e | S ^ S ^ I = |G 1| S - 5 _ ^ x 3 a > a 2 x 3 S 6 ' x —>-x— =>a x' — a _ x Thus S c CT S c = X € D. 5 5 s5a 2 3 v 6 s 5 3„l8 3 x 2L_> a x^ ±—-3-cta x a a . x^ 19 18 3 / 6 6 ) 5 = a a x^ » a ( a a x' / a 3 a 3 0 Hence CTY =» py where u » " \ € D. Th e r e f o r e S g G ^ » S 5 D S 5 + S 5 D Y S 5 C D + Dy - G x . Th e r e f o r e S k G x S k 'a. G 1 . k » 5 , 4 , 5 . 2.12 D e f i n i t i o n . G K = G k - 1 + G ^ S ^ ^ k = 3 , 4 , 5-22. 2.13 Theorem• i ) G k I s a group and I s k - f o l d t r a n s i t i v e on T k k =E 1, .; . f 5. i i ) G k S j V - S j G k k - . l , J-2 j =» 3, 5 i i i ) Gfc'5* {a's a € G^ and a f i x e s q ^ ^ * q^} k - s a-1, 4 Iv) Gg =• G-^  4- G^SgG-^. Proof. G 2 Is.doubly t r a n s i t i v e on T 2 by P r o p o s i t i o n 2.8. G 1 i s t r a n s i t i v e on • and G 2=» G-j^  + G 1S 2G 1 by lemma 1.2. \ * P k ( q k - l ' q k ) a n d P k f i x e s —•> ^5 f o r each k a 2, . . 5 by d e f i n i t i o n 2.1 and S f c 2 « 1 k=2, . . .>5j ( S k a l S k ) 3 » 1 k = 3, 4 , 5 and ( S m S k ) 2 » l m=2,...,k-2 k=»4, 5, (by P r o p o s i t i o n 2.10) S k G l S k " G l k - 3 * - V 5 (by P r o p o s i t i o n 2.11) Gx• - {CT S CT e G 2 CT f i x e s q 2) (by d e f i n i t i o n 2.6) H =» {CT S CT € G 2 a f i x e s q^ and q 2 ) . Therefore by the t r a n s i t i v e extension theory (theorem 1.3), G k i s a group and i s k - f o l d thransltive,'.on T k k=«3, 4 , 5 , and G k S j =» S^Gk k»l, j - 2 , j=3, 4 , 5 , and G k =» {CT ; CT e G^, CT f i x e s ^fc+i* ^5} k = l , ..., 4 . 2.14 D e f i n i t i o n . M-^ » G^ = the Mathieu group on 11 l e t t e r s . M 1 2 =» G^ > the Mathieu group on 12 l e t t e r s . 23. h o Theorem. JM^ J = 7920 .« 11-10 ?9>8 = 2 • 3 . 5'H |M12I - 95,040 = 12.|M n| - 2 6 . 3 5 ' 5 * 1 1 . P r o o f . M 1 1 i s 4 - f o l d t r a n s i t i v e on 11 l e t t e r s and H =» ,{x : x s ^l3 x f i x e s \> = {!}• T h e r e f o r e , hy lemma 1.4, |M1;L| = 1 1 » 1 0 ' 9 ? 8 ? 1 . S i m i l a r l y |M 1 2J » 12'11«10«.9<8«1.. 2k. S e c t i o n 3. I n v o l u t i o n s Of M. 11 and M 12* I n t h i s s e c t i o n , the i n v o l u t i o n s (elements o f or d e r 2) w i l l be c l a s s i f i e d a c c o r d i n g t o the number o f l e t t e r each f i x e s . The number o f i n v o l u t i o n s o f each type w i l l be c a l c u l a t e d and the r e l a t i o n s h i p between the 2-Sylow subgroups and the c e n t r a l -i z e r s o f the i n v o l u t i o n s w i l l be examined. The main r e s u l t s are. Theorems 3.6, 3.7 and 3.8. 3.1 D e f i n i t i o n , I T , T ^ , jy T,-, i y e M 1 2 are d e f i n e d by: IT : x—j»-x and IT f i x e s 0, «, v, w. k -1 Tk 5 x ~ > a x ' 0<~^"x>> v-ow f o r k=*l, 3, 5, 7. 3.2 Lemma I) A l l the" i n v o l u t i o n s "of => K l i e i n Q. i i ) I f CT i s an i n v o l u t i o n o f then CT f i x e s e x a c t l y 2 l e t t e r s I n T^ and cr i s conjugate i n G^ to I T . i i i ) 7T i s the o n l y i n v o l u t i o n o f G^ which f i x e s 0 and eo. ' ax + by cx + dy ^ \ { c 5 x 3 + d 3 y 3 } / . 3 [ ] 5 + a b 5 { } 3\ '(ab 5c - a 4 ) x + (ab^d - a^b)y (a^ac^+ cd^)x + (a\c3+ d 4 ) y Ax ^ By Lx •+ Ity Assume a 2 =» ( \ y ) 2 » 1 . Then Y A x + B y \ f o r every y, xeGF(9) \ / \ Lx + My/ y a n d x n o t b Q t h z e r o I n p a r t i c u l a r , i f x =» 0 and y » 1, y ^  so B = 0 . I f y =» 0 and x => 1, 0 \ ( B 5 - U A T h e r e f o r e ( *\ = ( j j * ) so A - M, so L =« 0 . B « 0 i m p l i e s a b 5 d - a^b « 0 3 3 3 L a 0 i m p l i e s a a c ^ + c d ^ =» 0 A =» M I m p l i e s ab 5 c - a 4 = o A c 5 + d 4 a 6 Q i m p l i e s d e t a =» 1 so ad - be = 1 Case I . Assume c = 0 o r b sa o E q u a t i o n (4) i m p l i e s ad = 1, t h e r e f o r e d = a " 1 h 4 -1 4 -4 4 E q u a t i o n (3) i m p l i e s -a =»d =» (a ) =» a = a . Th e r e f o r e -1 =»' 1 which i s a c o n t r a d i c t i o n . Hence c 0 and d o . 2 3 Th e r e f o r e e q u a t i o n (1) i m p l i e s cxb d - a. *> 0 ; 3 2 3 and eq u a t i o n (2) i m p l i e s a ac + d^ = 0 26. Case ,11'. Assume a =» 0 . E q u a t i o n (6) i m p l i e s cK '« 0 . Hence d = > 0 . Equation ( 4 ) i m p l i e s -be =» 1 so c a - b - 1 . E q u a t i o n ( 3 ) i m p l i e s - a b ^ " 1 = - a ^ b b - 3 . 2 3. - 2 4 2 Thus ab a a'ID and so b •=• a . 8 4 T h e r e f o r e 1 = b =» a = -1 which i s a contradiction. Hence a / 0 . Case I I I . Assume d = 0 E q u a t i o n (6) i m p l i e s a =» 0 which i s i m p o s s i b l e by case I I . Case IV. Assume a ^ 0 , b / 0 , c ^ 0 , d ^ 0 . 2 2 "*5 3 2 E q u a t i o n ( 5 ) i m p l i e s b c d + a a^c = 0 . "5" "3 2 2 ""5 2 2 2 "5 E q u a t i o n (6) i m p l i e s c r a ' x + a d p =« 0 . Hence b c d a a d , so b 2 e 2 a a 2 d 2 « ( a d ) 2 » (1+bc) 2 - l - b c + b 2 c 2 (by (4)). Hence be = 1 so c — b"" 1. —1 Thus e q u a t i o n (4) i m p l i e s ad - be a ad - bb = ad - 1 = 1. Hence ad =« -1 so d a - a " 1 . E q u a t i o n ( 3 ) i m p l i e s ab-^b - 1 - a 4 a a^bb" 3 + a - 4 so a b 2 - a^b""2 = a 4 + a""4 a 4 4 4 4 2 P . 4 4 6 . - 4 a + a =» -a . T h e r e f o r e 1 a (-a ) a cc D + a + a b - b 4 ( a 2 + a 6) + a 4 a a 4 '= -1. i . e . 1 » -1 which i s a c o n t r a c t i o n . 2 2 T h e r e f o r e the assumption t h a t CT a (\y) =» 1 i s f a l s e so t h e r e a r e no i n v o l u t i o n s i n Q y B u - b G^ = K a Q + Qy. Th e r e f o r e a l l the i n v o l u t i o n s o f G, l i e i n Q. L e t a e G, be an i n v o l u t i o n . Assume CT i s r e g u l a r 27. on ( i . e . f i x e s no l e t t e r s i n T ^ ) . a i n t e r c h a n g e s the l e t t e r s o f T^ i n p a i r s . Choose x, y e such t h a t x< 0 >j i s doubly t r a n s i t i v e on T^. Thus t h e r e e x i s t s X € G^ X X -1 2 such, t h a t x — ^ 0 , y- ><°. L e t p a X CT X G G^. U » 1 so |i s .Q^  0«s-^>°°. Hence |ja P ^ f o r s o m e 13 -€ G F ( 9 ) b / 0. But I " " I — ^ > ' a2 ^ jW b \ /b \ /a~^ b so i s f i x e d by |a. L e t z e T^ be such t h a t z — ^ a ^ b . Then .z i s f i x e d by a c o n t r a r y t o the choice of CT as a r e g u l a r i n v o l u t i o n . ; Then t h e r e are no r e g u l a r i n v o l u t i o n s i n G^ so CT f i x e s a t l e a s t 2 p o i n t s . L e t x, y be f i x e d p o i n t s o f cr. There e x i s t s X € G^ such t h a t x — ^ - 0 , y — ^ - 3 * ° ° . L e t - 1 2 M . « X CT X , M => 1, n s and \1 f i x e s 0, °°. T h e r e f o r e M Gi 3 D+Dy. Hence n = p o r n a py f o r p = f a 0 ,\s D. • x ' VO a " 1 ; Assume u a pY. p 2 v 6 3 P Y 6# ' 6 3x 3 4 24 x—-->a x — L ^ a a x ^ — F T >aa (aa x^) v a a a x a -x. 2 2 T h e r e f o r e x — ^ — > - x c o n t r a r y t o n a 1. Thus the assumption t h a t n a py i s f a l s e , so u a p. p 2 p 4 x "—>a x K > a x. 2 2 4 2 But u » p a l i m p l i e s a a l , i . e . a a +l or + a . Hence n.. a ( Q ° 1 » 1 o r | i / 2 o \ 2 But 1 i s not an i n v o l u t i o n . Hence ( i a p a a a y j \0 a ] (see Prop.. 2.4), and t h i s i s the o n l y i n v o l u t i o n o f G^. 28. \i j x—>~x f i x e s 0 , <» (and v, w) 3 so u' =• ir. But 0. and « a r e the o n l y f i x e d l e t t e r s o f TT i n T^. T h e r e f o r e x and y are the o n l y f i x e d l e t t e r s i n T^ o f CT. Hence i f a i s an i n v o l u t i o n o f G^, a f i x e s e x a c t l y 2 l e t t e r s and a i s conjugate i n 6^ t o T . ^•5 Theorem. I f a i s an i n v o l u t i o n o f M 1 2 » then i ) CT i s r e g u l a r and a ~ T f c f o r some k = l , 3, 5* 7 or i i ) CT f i x e s 4 l e t t e r s o f T^ and CT ^ TT. P r o o f . i ) Assume CT f i x e s no l e t t e r s o f T^ ( i . e . CT i s r e g u l a r ) . T h e r e f o r e CT exchanges the l e t t e r s o f T,- i n p a i r s . M L e t r.jV r ^ he such t h a t r^< g > r g J r ^ < ° >'r^ -. ^2 i s 4 - f o l d t r a n s i t i v e on T^, so th e r e e x i s t s X e M 1 2 such t h a t ^ — ^ 0 r 2 — ° ° r ^ >v, r ^ — ^ w . L e t „ 1 T T T » X CT X € Gp.. T f i x e s no p o i n t s and (X >° J 3 V > w . T £ G^ s i n c e T does not f i x w. Hence T a g^S^gg € G^S^G^, w g l > w s 5 > v & 2 >v. Hence g 2 f i x e s v. v s l >x— s-$-»y s 2 >w, y a w s i n c e g 2 f i x e s w. Hence x a v end g 1 f i x e s v. T h e r e f o r e g ^ g 2 6 G^. Thus" T € G^Sj-G^ « G^G^S^ a G ^ S ^ ( Q + Q Y ) S 5 - Hence T a gS^ or T a gyS,^ g e Q. and y f i x 0 and ». Hence 0<r-^-s6 g a ° f o r s o m e b e G P ( 9 ) W<>. Assume T a gS,_. z - A * - b V - M W - b 6 x ^ g S 5 , - b 6 ( - b V ) • - b V 29. 2 2 4 p But T a (gS,_) .'» 1 so b « 1. Hence b =» + 1 or b « + a . I f b > + l then a — g S 5 > -b^a""-5 =* -la"- 5 » a. Hence a i s f i x e d by T contrary t o the c o n s t r u c t i o n of T. I f b =»' + a2 then l - ^ > - b 6 ( l ) ~ 1 «* - ( a 2 ) » - a 1 2 a - a 4 * 1, Hence 1 i s f i x e d by T contrary t o c o n s t r u c t i o n of T. Thus the assumption that T € QS C i s f a l s e . Therefore T » gyS5 € Q YS 5 x g ^ . b 2 x - l V > -qb 6x - 3 S 5 > ^ 2 x " 1 a a 7b V 1 . By c o n s t r u c t i o n , T I s a r e g u l a r i n v o l u t i o n f o r any b s GF(9) b ^ 0. Hence there are e i g h t d i s t i n c t choices of b. But ( a 4 + n ) 2 a a ^ a . o 2 1 1 » ( a 1 1 ) 2 , so there are but f o u r d i s t i n c t 2 choices f o r b . Hence there are f o u r d i s t n c t choices f o r T ; T l 5 7 2 —1 x—*»a (a) x « « 7 ( a 5 ) V 1 a ax"*1 X --W^a 2) 2*" 1 a a 3 -1 a x T 5 ; X - ->a 7(a 3) 2x"" 1 a . a V ) 2 * " 1 a T 7 •: X - 7/ 4 , 2 -1 ->a (a ) x = ac a V 1 These are the o n l y r e g u l a r I n v o l u t i o n s Interchanging 0 and «°, v and w. Hence a i s conjugate t o one of them. i i ) L e t cr be an i n v o l u t i o n of M 1 2 f i x i n g 2 l e t t e r s x, y e Tp.. There e x i s t s X. e M 1 2 such t h a t x — v , y — -1 since M 1 2 i s doubly t r a n s i t i v e on T^. Hence X CT X f i x e s - I -1 v, w so X CT X 6. and i s an i n v o l u t i o n . Thus X CT X I s conjugate t o ir by lemma 3 . 2 . But ir f i x e s e x a c t l y f o u r l e t t e r s of T,_. Hence CT f i x e s e x a c t l y f o u r l e t t e r s of T^. 50. Assume p i s an i n v o l u t i o n of M 1 2 . f i x i n g f i v e l e t t e r s r^j r,_.. Then by 5 - f o l d t r a n s i t i v i t y , there e x i s t s X -1 X € M 1 2 such that r j • ><ly O3"1* 5- Hence X p\ e H 1 a {!}. i . e . X p X « 1, so p =» 1 and 1 I s not an i n v o l u t i o n . Therefore each i r r e g u l a r i n v o l u t i o n of M 1 2 f i x e s e x a c t l y f o u r l e t t e r s . P r o p o s i t i o n . T X , Ty and Tj of theorem 5 . 3 are conjugate, Proof. For each i«l, 5 , 5* 7, interchanges 0 and co ; v and w. S.~ interchanges v and w and f i x e s D -1 0 and « while y f i x e s 0, <», v, w. Hence y r^y9 -1 -1 S, - T , S c . and S N T C S c each interchanges 0 and °° ; v D -L D D D D and w. Y " 1 5 T l -1 - 5 - 5 Y -9 5 -1 x 1 — > . - a x y — y -aa x • =» -x i->.-ax ^ » a x -1 5 T l _ , - 5 S 5 _ „ 3 -9 _ J>v-l x—-—»».x s»ax ^—*. a x => a x S 5 1 3 T 5 5 - 5 S 5 1 5 - 9 7 -1 x —+—=?» x ^—5»a^x > a x ^ =« a x -1 -1 Therefore y T - J Y » T,_ ? S,- i ^ S j . =» and S 5 " " 1 T 5 S 5 . S 5 " 1 ( Y " 1 T 1 Y ) S 5 = ( Y S 5 ) ~ 1 T 1 ( Y S 5 ) - T ? . . D e f i n i t i o n . I f a € M 1 2, C(CT) - C e n t r a l ! z e r i n M 1 2 of {a}. I f a e M 1 1 ( CJ(CT) =» C e n t r a l i z e r I n M±1 of {a}. I f G i s a group then Z(G) s Centre of G. Theorem. I) There are 596 r e g u l a r i n v o l u t i o n s i n M 1 2 i i ) A l l the r e g u l a r i n v o l u t i o n s of M 1 2 are conjugate 31. i i i ) I f T i s a r e g u l a r i n v o l u t i o n s of M^ 2 then | C ( T ) ' | - 24o and T '£ Z ( P ) f o r every 2-Sylow subgroup P of M 1 2. Proof. i ) L e t TJ a [a i CT i s a r e g u l a r i n v o l u t i o n of M 1 2 ) . R - C(0;,r 2) ( r 5 , r 4 ) : r^TyT^ € T^, 0 £ £ £ r ^ £ 0, r ^ =» v or r 2 =» v and r ^ =» «>}. Let (0, r 2 ) (Ty r^) e R. I f r ^ a v then there are 10 d i s t i n c t p o s s i b l e values of r 2 and f o r each of these there are 9 p o s s i b l e values of r ^ . Thus there are 10x9 - 90 choices of ( 0,.T 2)(v, r^) € R. I f r 2 » v and r ^ = « then there are 9 p o s s i b l e values of r ^ . Thus there are 9 choices.of (0, v) (<*>, r^) Hence JRJ - : 9 0 + 9 =* 99-For each (0, r 2 ) (ry r^) e R there i s a X e M 1 2 •x \ x x such t h a t 0—^-*0 ? r 2 — ^ « ^ r ^ — % - v and r ^ >w. Le t cri = X T ^ X " 1 , i - 1 , 3, 5, 7 . i s a r e g u l a r i n v o l u t i o n and 0 < CTi > r 2 , r ^ < CTi ^  r ^ . I f p. 6 U and 0* > r g , Hence r^<.^ >r^ then l e t T = X" p.X. 0•<•T> «>, v< T>w T a f o r some i = l , 3 , 5, 1, (by Theorem 3 - 3 ) , so u a f o r some i . Therefore f o r each (0, r 2 ) ( r ^ , r ^ ) e R there are e x a c t l y f o u r r e g u l a r i n v o l u t i o n s CT^ such t h a t <J« CTJ 0 <, > r g r ^ - e — ^ r ^ . L e t cr € U . CT, i s a product of s i x t r a n s p o s i t i o n s and 0 must appear i n one of these. I f o < ° >v then « must appear i n another t r a n s -32. p o s i t i o n and a =• ( 0 , v) (<*>, r^) (r,_, r 6 ) # * * ( r n > r l 2 ^ * I f 0 <• g > r g ^ v then v must appear i n some other t r a n s p o s i t i o n and CT =» ( 0 , r g ) (v, r^) ( r ^ , rg) ... ( r 1 1 , r 1 2 ) . Hence f o r each CT e U t h e r e e x i s t s (0j,r 2) ( r ^ , r ^ ) € R such t h a t CT - ( 0 , r 2 ) {Ty r^) ( r 5 , r g ) ... ( ^ p r ^ ) . T h e r e f o r e |UJ 4|H | => 4x99• - 396. i i ) L e t T he a r e g u l a r i n v o l u t i o n o f M ^ 2 . T i s conjugate t o f o r some i=»l, J>, 5* 7 (theorem 3.3) and the T ^ ' S are conjugate. Hence a l l the r e g u l a r - i n v o l u t i o n s o f M^ 2 a r e conjugate. i i i ) The number o f conjugates o f T i n M 1 2 =* [ M 1 2 : C ( T ) ] =» the number of r e g u l a r i n v o l u t i o n s o f =• 396. T h e r e f o r e j G( T) \. - i ^ L . 1 2 x 1 1 x 1 0 x9x8 m 2 k Q 396 4x11x9 4 2 x3x5. Assume T € Z ( P ) f o r some 2-Sylow subgroup P Of M 1 2 - Then | P | = 2 ^ . T h e r e f o r e T X =« X T f o r each x e P so P C C ( T ) . C ( T ) | i . e . 2 6 4 2 x3x5, which i s a c o n t r a d i c t i o n . T h e r e f o r e |P T h e r e f o r e T £ Z(P) f o r every 2-Sylow subgroup P of M 1 2-3.7 Theorem. i ) A l l the i r r e g u l a r i n v o l u t i o n s o f M 1 2 a r e conjugate. i i . ) I f a i s an i r r e g u l a r i n v o l u t i o n o f M 1 2 then |C(CT)| = 192 and t h e r e e x i s t s a 2-Sylow subgroup P o f M 1 2 such t h a t CT e Z ( P ) . i i i ) I f V i s a 2-Sylow subgroup o f M 1 2 then the^e 33. e x i s t s an I r r e g u l a r i n v o l u t i o n X o f M 1 2 such t h a t X e Z ( V ) . Pr o o f i A l l the I r r e g u l a r i n v o l u t i o n s o f M 1 2 are conjugate "by Theorem 3.4. 1 : L e t R =» {0, eo, v, w} =* the set of l e t t e r s f i x e d by i r . L e t X e .C(ir) . T h e r e f o r e Xi r •«* T r X and X ^ i r X =» i r . I f r e R then r x » ( r ^ - r F X - r X 7 r •- ( r X ) 7 r . . Hence r X I s f i x e d by i r . So , r X e R. I f |i i s an i n v o l u t i o n o f M, 0 and r ^ G R f o r each _ -1 „ -1 •• Lsi -1 r e R then r w - ( r ^ ) 7 ^ =- ( r ^ '«. ( r ^ ) ^ . - r . Hence liTTia-"1 f i x e s each r G R. T h e r e f o r e utT^r 1 = i r s i n c e i r i s the'.only I n v o l u t i o n o f M 1 2 f i x i n g each r G R. Th e r e f o r e u 6 C ( i r ) . T h e r e f o r e a e C ( i r ) i f f r C T 6 R f o r each r e R. (*) Define cp : G ( i r ) — ^ ^ 2 1 . (Symmetric group) by cp(a) . (° v w \ 0 a » a v a w°/ cp i s a homomOrphism and cp i s onto because: / 0 » v w ] _ I f x a> I b c a e ^ then R =» {a,b,c,d}. I s some o r d e r i n g o f R (ordered s e t s ) . There e x i s t s a G M^ 2 such t h a t O g>-a, °° CT b v — — > c , w—a->>d and a 6,C ( i r ) by (*) . Hence cp(a) = x. Th e r e f o r e f o r each x e s& ^  t h e r e i s a a':e;C(ir) such t h a t cp(a) =» x. : K e r n e l o f cp =» [a : a G.C ( i r ) and cp(a) '=» 1} =• {<? 1 a G G ( i r ) and a f i x e s 0, <*>, v, w) c G, 3 * . But G1 c C(ir) by (*) . Hence • k e r n e l o f cp. T h e r e f o r e C(Tr)_/G^ ~ ^ f ^ . Thus | G(ir)j » |G1| x j ^ f ^ l » 8x4l » 192 « 2 6 x 3 -The number o f i r r e g u l a r i n v o l u t i o n s o f M 1 2 » numbers o f conjugates i n o f TT >• [ M 1 2 : C(TT) ] 12x11x10x9x8 a 8x4! L e t a be any i r r e g u l a r I n v o l u t i o n o f M^ 2. Then 495 = the number o f I r r e g u l a r i n v o l u t i o n s o f M 1 2 «• the numbers o f conjugates I n M^ 2 t o a - [ M 1 2 : C(.o)]. Hence | C(a) | =» 192 - 2 6 x 3 . L e t P be, a 2-Sylow subgroup o f C ( a ) . |P|' 2 so P i s a 2-Sylow subgroup o f M 1 2 « CT e x " 1 Px f o r some x £ C(CT) s i n c e cr must l i e i n some 2-Sylow subgroup o f C(a) and a l l these are conjugate i n C ( a ) . Hence x CT x " 1 e P. But x CT X" 1 » CT s i n c e x e 0 ( a ) , so CT e P. T h e r e f o r e cr i s contained I n each 2-Sylow subgroup o f G(CT). A l s o y a > ay f o r each y e P s i n c e P c C ( a ) , SO a e Z ( P ) . L e t V be a 2-Sylow subgroup.of M^g. V i s a p-group. Hence Z(V) c o n t a i n s an element b e s i d e s the i d e n t i t y . ' . i . e . Z(Y) c o n t a i n s an i n v o l u t i o n X, and V c C(x) . X i s not r e g u l a r . (Theorem 3 . 6 ) . So X i s a n . I r r e g u l a r i n v o l u t i o n . Hence ¥ c C(x) f o r s(ome i r r e g u l a r i n v o l u t i o n X. 35. 3 .8 Theorem, i ) There are 165 i n v o l u t i o n s i n M.^. i i ) I f a i s an i n v o l u t i o n o f then CT f i x e s 3 l e t t e r s . o f T^ and a i s conjugate i n to ir and a e Z(P) f o r some 2-Sylow subgroup P o f I i i ) I f V i s a 2-Sylow subgroup o f P then t h e r e e x i s t s an i n v o l u t i o n X o f '^2.1 s t l c h t h a t X e Z ( V ) . P r o o f . L e t CT be an i n v o l u t i o n o f M ^ . CT e i m p l i e s CT f i x e s w =» q^ so CT f i x e s some y € T^. There e x i s t s X e such t h a t y — ^ v s i n c e i s t r a n s i t i v e on T^. L e t \i •=» X^CTXeM^. u I s an i n v o l u t i o n f i x i n g v (and w) so u e =• K. Hence u i s conjugate i n t o ir. T h e r e f o r e CT i s conjugate I n t o ir and CT f i x e s e x a c t l y 3 p o i n t s o f T^ s i n c e ir f i x e s 3 p o i n t s o f T^. L e t R =» {0, v, «>} s e t o f f i x e d l e t t e r s o f ir i n . 1 • • T^. 1 As i n theorem 3 . 7 , x e fJ(ir) i f f r x e R f o r each r e R. D e f i n e cp : (T(ir)—Zed-, (symmetric group). As i n theorem 3 . 7 , cp i s a homomorphism onto and G^ =» k e r n e l o f cp. T h e r e f o r e ^(v)/^ ~ ^ y and |?(ir) | =- | G 1| x| - 8 x 3 ! - 4 8 . - 2 4 x 3 . The number o f con^jiggates I n o f ir =» [M^tC^ir) ] .Ox 11x10 9x8 l 6 5 36. m the number o f i n v o l u t i o n o f M^^ -=• the number o f conjugates o f g [H u : ? T(a) ]. Hence |C"(a>-|. - 48..- 2%5'. , L e t P be a 2-Sylow subgroup o f (T( cr) . Then |P| = 2 4 and P i s a 2-Sylpw subgroup o f M^. As i n theorem 3 . 7 , cr e Z ( P ) . L e t V be any 2-Sylow subgroup o f ML^ '. As I n theorem 3 . 7 , t h e r e e x i s t s an i n v o l u t i o n u e Z ( V ) . 37. S e c t i o n 4. C o n s t r u c t i o n o f M 2 2, Mg^, M 2^ D e f i n i t i o n , i ) Q > PSL(3,^) •'. - SL(3,4)/Z where Z » c e n t r e o f SL(3,4). i i ) T 2 - fx t x = ( x ^ x ^ x ^ ) , x ± e GF(4), x j& (0,0,0)}. F o r x € T 2 , x"'• (x-^x^x-j) =» ( a x ^ a x ^ a x ^ ) =» ax f o r any a e GF(4)^ a / 0. 1 j i i i ) q x'.- (0,1,0) € T 2 , q 2 - (1,0,0) € Tg. l y q i ^ ar e d i s t i n c t l e t t e r s hot i n Tn. hy: \ - T 2 U {q 3, ...,q k) 'k - 3, . 5 V T 2 " { q 2 3  T o •- T 2 -Iv) p i s a p r i m i t i v e element o f GF(4). P + 1 - p 2 ) . v) S 2, S^, S^, S^ are permutations on T,. d e f i n e d ( x ; y , z ) — > ( y , x , z ) f i x e d (x,y,z) > ( x 2 + y z , y 2 , z 2 ) q 2 ^ _ > q 5 , q^,q 5 f i x e d . (x,y,z) =>(x 2,y 2,pz 2) q ^ ^ - ^ q ^ (x,y,z) > ( x 2 , y 2 , z 2 ) q4<->q 5 q^ f i x e d q^ f i x e d P r o p o s i t i o n . Z - {I, p i , p 2!} where I - i d e n t i t y o f S L ( 3 ^ ) , and thus (a^j)•'«• (pa^ .,) »(p a^.,) f o r any (a., ^ ) € PSL(3,^). P r o o f , g e Z i f f CT.=» a l f o r some a e GF(4), such t h a t .3 , . « «2 a";» 1. Hence a = 1> p o r p , so g =• I , p i o r T h e r e f o r e Z = {I, p i , p 2 ! } . So I n PSLp,*), ( a ^ ) .2. Noter I f ( a ^ ) e Q » PSL ( 3 , 4 ) ancl / 0 , then ( a ^ ) Hence b,, =• 1. ( h i d ) w h e r e ^ . - a ^ . — "11 I.e. Any elemient o f Q can he represented as m a t r i x V°^) w i t h b.^ -«• 0 o r 1. D e f i n i t i o n . G 2 « Q =» PSL ( 3 , 4 ) G^ =» {cr : CT e G 2 and a f i x e s q 2 ) H =» {a : a e G 2 and f i x e s q 1 and q 2 ) P r o p o s i t i o n . G^ H CT : a € Gg, CT "a : a € G«, a P r o o f . L e t cr =» € G p. Then q r S i m i l a r l y q, ' a l l a 1 2 a 1 3 a 2 1 a 2 2 a 2 5 a 3 1 a 3 2 a 3 3 ( 0 , 1, 0) l 21 l 31; ( a 1 2 j a 2 2 , a^ 2) a e G, i f f o* f i x e s q, i f f ( a n , a 1 2 , a 1 3 ) - ( 1 , 0 , 0) i f f a 12 l 13 CT € H i f f CT € G 1 and cr f i x e s q^ I f f CT e and ( a 1 2 , a 2 2 , a ^ ) » ( o , 1, 0) 0 . I f f CT € G^ and a 1 2 "32 39. 4 . 6 P r o p o s i t i o n . i s t r a n s i t i v e on and Gg i s doubly-t r a n s i t i v e on Tg. P r o o f . L e t x = ( x 1 , x^) e I f x 0 ^ 0 l e t X » 1 x 1 x - 1 0^  x e G, Then x 3 x2,y + x^ ,XgX^+XgX— I f x 2 » 0 then Xy ji 0 because {x^,0,0) » q 2 ^ and (0,0,0) / L e t X Then x a , x 1 + x 1 1 0 q 1 . Hence G 1 i s t r a n s i t i v e on T n . I n each case x L e t x a (x^XgjX^): e T 2 . I f x^ ^eOClet X « 0'' 0' Then•x X X ^ X g + X 1 X g 40. I f x, 0 arid x n d 0 l e t X Then x =» 0 x^ 1 x 2 °\ 0 0 *3 r ,1 0 , o y 0 x 0 2 1 6 a 2 ' \ X 2 X 3 + X 2 X 3 I f x^ s» o and x 2 s* 0 theii x^ ^  0 . L e t X =» -1\ Then x 0 0 x^ 1 0 0 0 x^ 0 /o 0 x; 1 0 o ' \0 x, 0 £ G Therefore, f o r any x e Tg there e x i s t s X e G 2 such that. x X =» ( 1 , 0 , 0) » q 2 . Hence G 2 i s t r a n s i t i v e on T g. So G 2 i s doubly t r a n s i t i v e on Tg sin c e G.^  i s t r a n s i t i v e on T^. (Lemma 1.3) P r o p o s i t i o n . .S k' .1 e H 3 _ ( s n s k ) 2 • k - 2, 3, 4 , 5 k - 3, 4, 5 n m 2,.. .,k~2, k =« 4 , 5 . Proof. S k ^ ( l ^ - l ^ ^ k ^ where P k i s a permutation f i x i n g q ^ , . . . , f o r k => 2 , . . . , 5 . S I S k 5 8 P i ( % . l ^ ) P k ( q k - l ^ k ) k2- . i f i . =- k P k - l P k ^ k - 2 ^ k ^ k - l ) i f 1 " k " 1  Pn Pk( qn»l^n) ( W k * l f 1 " n ^ k ~ 2 * 41, Hence S k 2 a P k 2 k * 2 , 5 ( S k ~ l s k ) 5 - ( p k - l P k ) 5 4, 5 ( S ^ f » ( P n P k ) 2 n-2, k-2 ka4, .5. Hence i n the f o l l o w i n g we need not c o n s i d e r q ^ , q ^ . (x,y,z) (y,x,z) ^ - ^ ( x , y , z ) ( x 3 y , z ) " 3 >(x 2+yz, y 2 , z 2 ) - ^ > ( ( x 2 + y z ) 2 + y 2 z 2 , y \ z 4 ) « ( x + y 2 z 2 + y 2 z 2 , y, z) a (x,y,z) 2 2 «_2s b 4 ^ , 4 4 ( x , y , z ) ; ^ ( x * , y * , 0z^) ^ > ( x ^ , y \ 0 ( i 3 z 2 ) 2 ) - (x,y,z) p • p p 4 4 4 ( x , y , z ) 2 - ^ ( x % y % z ^ — Z ^ x , y , z ) - ( x , y , z ) Hence Sy.2 = 1 k - 2, 3, 4, 5. S S I f z f 0, (x,y,z) « — ^ ( y j X j Z ) ^U-(y 2+xz, x 2 , z 2 ) S2 S3. x 4 + ( y 2 + x z ) z 2 , ( y 2 + x z ) 2 , z 4 ) o p 2 2 x+y z +x, y+x z , z) o p p 2 y z , y+x z , z) ( y + x 2 z 2 ) 2 + y 2 z 2 z , ( y 2 z 2 ) 2 , z 2 ) 2 2 2N y +xz+y , y z , z ) xz, y z , z 2 ) a (x,y,z) I f z-0> ( x , y , 0 ) — — 2 - > ( y , x , 0 ) — ( y 2 , x 2 , 0) S 2 S 3 But (x,y,0) e T . Hence x f 0 and y ^ 0, So ( y 2 , x 2 , 0) a (xy-y*, xy-x, xy-0) = (x,y,0).. 3 Thus (x,y,0) *- J >(x,J,0) and so (x,y,0) 4*(x.,y,0) T h e r e f o r e ( S g S ^ ) 5 * 1 ( x , y , z ) " 5»(x 2+yz, y 2 , z 2 ) — t > ( x + y 2 z 2 , y, pz) S ? S 4 > ( x+y 2 z 2 + y 2 p 2 z 2 , y, p (pz) ) a ( x + ( y 2 z 2 ) ( l + p 2 ) , y, p 2 z ) - ( x + P y 2 z 2 , y, p 2 z ) S-zSj, p p p p % 2-2^.(x+ftrz* + y -pz , y, p^z) - (x,y,z) Hence (S-^S^) .=» 1. ( x , y , z ) — ^ > ( x 2 , y 2 , p z 2 ) - ^ ( x , y , p 2 z ) S S ^ - ( x , y, P 2 ( P 2 Z ) ) - (x> y, pz) 3 4 % ( x , y , p 2 ( P 2 z ) ) - (x, y, z) Hence ( S ^ S , - ) - 5 » 1. S 0 p p o (x,y,z) ^ > ( y , x 3 z) =-*-(y% x % pz ) S S 2 4> (x, y, P f p z 2 ) 2 ) » (x, y, z) T h e r e f o r e (SgS^) a l . So Sj- 2 2 2 (x,y,z) ^ ( y , x , z) 2_^(y• , x , z ) S,-vSp-— £ - 2 ^ > ( x , y, z) T h e r e f o r e (SgS^) 1 _ 5 ^ / „ 2 , „ 2 „2N °5 (x,y,z) =^(x + yz, y , z ) -*(x + y 2 z 2 , y, z) •^(x + y 2 z 2 + y 2 z 2 , y, z) - (x, y, z) . Therefore ( S ^ ) = 1 . Proposition. S^^S^ =• k => 3, 4, 5. Proof. Let g e G r S ^ - P k ( q k _ p q k ) g P ^ ^ q ^ => P k g Pfc since Pfc and g f i x If g z, then det.g =» cf + de =• 1 2 2 2"1 x +yz+ay +bz 2 . 2 _w 1 cy + dz 2 2 ey^ + f z ^ o 2 2 2 2 2 2 2 \ '(x +yz+ay +bzc) + (cy +dzc) (ey +f z ) 2 2 c y + d z 2 2 e > + f dz fx + (a2+ce)y + (b 2+df)z N 2 2 c y + d z 1 a2+ce b2+df e ^ + f 2 z Therefore S^gS^ - X 1 a2+ce b2+df ? ? 2 2 and det.Xac^f^+d e 2 (cf+de) 9 1 . Thus X e G x so S ^ S ^ c 2 2 2s x +ay +b0z 2 2 cy c+d0z e y 2 + f 0 z 2 / 2 2 2 2 2 2 c y+d 0 z \ 0e 2y+f V z , Therefore S^gS^•- X • 0e' h 2 0 2 S d 2 0 2 and det X » (cf+de) 2 » ] So X e G 1 and S^ G^ S^^  c G 1. 2 2 2\ x +ay +fcz 2 2 c y c + dz 2 2 ey + f z x+a 2y+b 2z 2 2 c y+d z 2 2 e y+f z Therefore 3,-gS,-5 5 •1 a 2 1,2' 0 e 2 d 2 ,0 e 2 f 2 and det X - c 2 f 2 + d 2 -. (df+de) 1 So X € G 1 and S^G^S^ c G-^  Therefore S kG 1S k c G x f o r each k = 3, 4, 5. D e f i n i t i o n . Gfc - G f c - 1 + k - 3, 4, 5. 45. "4.10 Theorem. i ) G k i s a group and i s k - f o l d t r a n s i t i v e on T k f o r k a 1, .. 5. i i ) G^Sjj, - j - 1, - k-2J- k - 3, 4, 5. I i i ) H S k - SfeH k m 2, . . . , 5 . i v ) Gfe => {a : a € G,_ and CT f i x e s q f c + 1 , . .. ,q^3 f o r k 2, 3 5 4. v) G 2 = G 1 + .G 1S 2G 1 Proof. G 1 i s t r a n s i t i v e on T^ and Gg I s doubly t r a n s i t i v e on Tg by p r o p o s i t i o n 4.6. Gg = G 1 + G 1SgG 1 by lemma 1.5. s k 2 - 1 e H, ( s ^ ) 3 - ! , ( w f 1 n < k~2 ^ p r o p o s i t i o n 4.7 and s j c G 1 S k = G-j_ k = 3, 4, 5 by p r o p o s i t i o n 4 . 8 . Therefore i ) , i i ) , i i i ) , i v ) f o l l o w by the t r a n s i t i v e e x t e n s i o n theorem (theorem 1.6). 4.11 D e f i n i t i o n . Mgg - Gy » Gjp Mg^ » Gy 4.12 Theorem. |M22| » 48 x 22 x 21 x 20 =» 2 7 x 3 2 x 5 x 7 x 11 i M 2 3 | m 23 x |M22| ..- 2 7 x 3 2 x 5 x 7 x 11 x 23 l M 2 4 l " 2 4 * l M 23l = 2 1 0 x 3 5 x 5 x 7 x 11 x 23 Proof. M 2 2 i s 3-fold t r a n s i t i v e on 22 l e t t e r s and ( H ss {CT ; CT e M 2 2, CT f i x e s q x , q 2 , q^} C [1 0 b = < a i a = I 0 c d L \0 0 e" by proposition 4.4 Thus |HJ = 4 x 4 x 3 ™ 48 since there are 4 possible values f o r bj 4 f o r d; and 3 f o r c. Therefore by lemma 1.4, JM 2 2( - 22' x 21 x 20 x 48. S i m i l a r l y , \M2^\ = 23 x 22 x 21 x 20 x 48 |M24| = 24 x 23x 22 x 21 x 20 x 48. b, c, d e G P(4) c'.f 0 47. S e c t i o n 5. I n v o l u t i o n s of M 2 2, M^, M^. I n t h i s s e c t i o n , p r o p e r t i e s analogous t o those o f s e c t i o n 3 a r e found f o r the i n v o l u t i o n s o f M 2 2, M^, and M^. A l l the r e s u l t s are s t a t e d I n the f o u r theorems (5.2, 5*4, 5.5, 5 . 6 ) , and then proved hy a s e r i e s of p r o p o s i t i o n s . 5.1 D e f i n i t i o n . For x e M^, C(x) = c e n t r a l i z e r i n M 2^ of {x}. Fo r x e M2^, TT(x) = c e n t r a l i z e r i n Mg^-of {xj. F o r x € M 2 2, C(x) =« c e n t r a l i z e r I n M 2 2 o f {x}. 5.2 Theorem i ) I f CT i s a r e g u l a r i n v o l u t i o n o f then CT i s conjugate i n t o sOme XS^S^ € G^S^S^ and t h e r e are 90 such X's. i i ) There are 43, 470 = 90 x 23 x 21 = 2 x 3 5 x 5 x 7 x 23 r e g u l a r i n v o l u t i o n s i n M2^. i i i ) I f CT I s a r e g u l a r i n v o l u t i o n o f M 2^ then CT £ L(P) f o r every 2-Sylow subgroup P o f Mg^. T h i s theorem i s proved i n P r o p o s i t i o n s 5«7 to 5.12. 5-3 Remark. T h i s theorem i s analogous to theorem 3.6 f o r M 1 2 except t h a t we were unable to determine whether o r not a l l the r e g u l a r i n v o l u t i o n s o f Mg^ are conjugate. As a r e s u l t , we cannot c a l c u l a t e | C ( c r ) | where CT i s a r e g u l a r i n v o l u t i o n . I n p r o p o s i t i o n 5.11 we show, th e r e are a t most e i g h t conjugate c l a s s e s o f r e g u l a r i n v o l u t i o n s . 48. Theorem. i ) I f a i s an i r r e g u l a r i n v o l u t i o n o f Mg^ t n e n a f i x e s e i g h t l e t t e r s of T,~ and CT i s conjugate to \x e H and u - (o i b I a ? b € GF(*)> a ^ G or b ^ 0 . \0 0 1, i i ) A l l the I r r e g u l a r i n v o l u t i o n s of Mg^ are conjugate i n Mg^. i i i ) There are 11,385 = 3 2 x 5 x 11 x 23 I r r e g u l a r i n v o l u t i o n s i n M2^. Iv) I f cr i s an i r r e g u l a r i n v o l u t i o n of Mg^ then |C ( a ) | = 21,504 . 2 1 0 x 3 x 7 and CT € Z(P) f o r some 2-Sylow subgroups P of Mgip This theorem i s proved i n P r o p o s i t i o n s 5.13 to 5 . 2 2 . Theorem, i ) There are 3795 = 3 x 5 x 7 x 2 3 i n v o l u t i o n s i n Mg-j and a l l are conjugate i n Mg^. i i ) I f a i s an i n v o l u t i o n of Mg-^  then a f i x e s seven l e t t e r s of T^ and J(T(CT)| - 2,688 = 2 7 x 3 x 7 and CT 6 Z(P) f o r some 2-Sylow subgroup P of Mg^. Theorem. I) There are 1155 = 3 x 5 x 7 x 1 1 i n v o l u t i o n s i n Mgg and a l l are conjugate I n Mgg. i i ) I f CT i s an i n v o l u t i o n of Mgg then CT f i x e s s i x l e t t e r s of T^ and | C ( C T ) | = 384 = 2 7x 3 and CT € Z(P) f o r some 2-Sylow subgroup P of Mgg. These two theorems are r e a l l y c o r o l l a r i e s to theorem 5.4 and are proved i n P r o p o s i t i o n s 5-23 to 5-27. 4 9 . 5.7 P r o p o s i t i o n . I f a i s a r e g u l a r i n v o l u t i o n o f M^^ then a ~ u e G-^SjS^. P r o o f . a Interchanges l e t t e r s o f i n p a i r s s i n c e i t f i x e s no l e t t e r s . L e t r 2 , r ^ 6 he such t h a t r 2 « ^ - > r ^ , r^<i-CT-^'r^. There e x i s t s X G M 2^ such t h a t r f c — ^ - ^ Q f c k=2, 5 s i n c e M 2^ i s 4-fold t r a n s i t i v e on T,_. L e t u =•' X~^ " CT X. q2-<. ^ > qi|."<-^ -> q^ and u 2 =• 1 so u £ G^ because elements o f G^ f i x q,-. T h e r e f o r e u - g S 5 h € G ^ S ^ . q 5 _ i ^ . q 5 ^ q ^ - J L ^ q ^ so h f i x e s q^ Hence h G G^ so u G G^S^G^ = G^G^S^ = G^S^. L e t ia => gS^ 6 G^S 5. q ^ — — T h e r e f o r e y • q^ . so g f i x e s q^. Thus :'..g. G G^. Hence \x = gS^ e- G^S^. • f i x e s q c and q^ and ^ = gS,- : q 2«* >q^. * T h e r e f o r e q g ^ ^ ^ q - ^ and g G G^. g £ Gg s i n c e elements o f G 2 f i x q^ so g - f S ^ h 6 G 2 S 5 G 2 , q ^ — L ^ q ^ ^ q 2 _JL^q 2 . Hence h f i x e s q 2 , so h G G.^  and g 6 GgS.^ = GgG^S^ => GgS^. f S 3 q 2 =>y ^>q^. T h e r e f o r e y = q 2 so f e G 1. T h e r e f o r e g G G^S^ SO U G G-^ S-^ S^ . 5.8 P r o p o s i t i o n . I f XS^S^ i s a r e g u l a r I n v o l u t i o n o f M 2^ then (1 a a 2 x \ j 1 a e\ 0 1 0 a ^ 0 o r X = 0 .1 0 , e f 0 0 0 1 / X s P ; o r ^ \0 e 1/ 50. or X a d f 0 or A a b 0 0 d d +" 0 \0 d 2 0/ b + a d .+ d ' o r PrOof. L e t \ = x \ /x+ay+bz y I — ° y + d z c =B p or p' b f ac +c S 5 S 5 € G 1* arid there are 90 such X's Then c f + de 2 2 V  r(x+ay+bz) + (cy+dz) (ey+fz) cy + dz ey + f z 3 5 . fx + (a+c 2 e 2 ) y + (b+d 2f 2)z + y 2 z 2 ' cy + dz ey + f z fU + (a+c 2e 2)V + (b+d2f2)¥ + v V ctV + dW eV + fW Therefore y = cV + d¥ = (c +de)y + d(c+f)z p and z = eV + fW = e(c+f)y + ( f +de)z These must be true f o r any values of y arid z I f y = 0 , z = 1 then (1) i m p l i e s d(c+f) = 0 and (2) i m p l i e s f +de = l 2. I f y sa l , z 0 then (1) i m p l i e s c +de » 1 and (2) i m p l i e s e(c+f) = 0 i f XS-S,. 3 5 i s an i n v o l u -t i o n . (1) (2) (not both zero) (3) (4) (5) (6) .2.2 (4) and (5) imply f +c - 0. Hence c»f. (3) and (6) are 51. s a t i s f i e d f o r any value of d, e. :: (4) i m p l i e s c«l i f f d=0 o r e»0 (7) x m U + (a+c 2e 2)V + (b+d 2c 2)W + ? V = [x + (a+c 2e 2)y + (b+d 2c 2)z + y 2 z 2 ] 2 2 2 2 + (a+c e )(cy+dz) + (b+d c )(ey+cz) + [ (cy+dz)(ey+cz)] 2 "5 2 2 2 = x + (a;+ ac + c^e + be + c d e)y p ~*> p + (b + ad + c de + be + c-^ d ) z . Therefore a. + ac + c^e + be + c d e =» 0 (8) 2 3 2 b + ad + c de + be + c^d = 0 (9) Case I . d * 0 and e = 0. Hence c »,1. Thus XS^S^ f i x e s (x,y,z) € T 1 i f f ay + bz + y 2 z 2 = 0. This equation has s o l u t i o n s i f a = 0 or b = 0 or 2 2 a = b = 1 or a, = b = B o r a = B , b = p. Otherwise I t has no s o l u t i o n s . Il a b\ Hence XS,S_ i s a r e g u l a r i n v o l u t i o n when X => 0 1 0 ^ 5 \0 0 1 2 2 and a = 1, b = B or b ; or a = B, b » 1 or B; or a » B b = 1' or B 2 . 2 2 2 i . e . when a «= 0 and b = Ba or B &-• Case I I . d = 0 and e ± 0. Hence c =* 1 . p (8) Implies e + be = 0 so b = e since e |= 0. (x + (a + e 2 ) y + ez + y 2 z 2 ey + z I f (x,y,z) i s fixed by XSy3^ then z = ey + z. Hence y =B 0 since e 4s 0. Thus x =« x + "ez so z = 0. But (x,0,0) = q 2 £ T^, so XS^S^ f i x e s no l e t t e r s of T 1. / l a e\ Therefore i f X =» 0 1 o j then XS^S^ Is a regular involu-\° e X ' t i o n ( i f 0 ) . Case I I I , e ss o, d j= 0. Hence c » 1 . (9) implies ad + d = 0 so a = d since d =}= 0. p p p\ /x + dy + (b+d )z + y z X S5 S5 > | y + dz I f (x,y,z) • i s fixed by XS^S^ then y = y+dz. Therefore z .» 0. Also y = 0 since x =» x + dy. But (x,0,0) = q 2 ^ T^y so XS^S^ f i x e s nO l e t t e r s of T^. /i a b\ Therefore, i f X = I 0 1 d and d 4s 0, then XS^S^ i s a \0 0 1 / regular i n v o l u t i o n . Case IV. d ± 0 and e f 0. Hence c ^ 1 . (5) implies e • d 2 ( l + c 2 ) , e 2 = d(l+c) (8) implies a + ac + c 5d(l+c) + bd 2(l+c 2) + c 2d 2d 2(l+c)=0 53. 3 2 2 T h e r e f o r e (1+c)[a + c d + (bd +c d ) ( l + c ) ] = 0, o p p but 1+c ± 0 so a + bd + c d + bed , =0. 2 2 2 2 Thus a + bd (1+e) + c d = 0, and so a = c d+bd (1+c). (9) i m p l i e s b + ad + c 2 d d 2 ( l + c 2 ) + be + c 5 d 2 » 0 b + c 2 d 2 + b ( l + c ) + c 2 ( l + c 2 ) + be + c 5 d 2 = 0 c 2 d 2 + c 2 + c + c 5 d 2 = 0 d 2 ( c 2 + c 5 ) + (c 2+c) =0. But d 2 ( c 2 + c 5 ) + ( c 2+c) = fo i f c d 2 c + l i f c = B o r p 2 . T h e r e f o r e XSy3^ i s an i n v o l u t i o n f o r any value o f d ^ 0 2 i f c » 0 and f o r d = c i f c » p or B • i ) i f c » 0 then e = d 2 2 2N x + ay + bz + y z dz d y I f (x,y,z) i s f i x e d by XS^S^ then y a dz. Hence 2 4 2 x » x + adz + bz + d z , so 0 a (ad + b + d ) z . I f z a 0 then y a 0. But (x,0,0) a q 2 £ T^. Hence z \> 0. T h e r e f o r e (x,dz,z) e T^ i s f i x e d by XS,S C i f f (ad+b+d 2) a 0. , 5 5 K / l a b ' T h e r e f o r e XS^S^ I s a r e g u l a r i n v o l u t i o n when X a I o 0 d and b ^ (a+d)d. AO d 0, 2 i i ) i f c a p o r p then d a c,. Hence e a l . 54. 2_2N /x + (a + c )y + (b + c ) z + y z XS-yS r-3 5 . cy + cz y y + cz I f (x,y,z) e T - ^ i s f i x e d by XS^S^ then y = cy + cz and P ' p P z = y + cz, i'.'-e.("y:•-•=• ^c.:z.;:- Thus..* x- =» x + ) (a+.c. ) c z-+ ::(b+c)z cz p p so 0 = (ac^+c+b+c+c)z ,= (ac +c+b)z. But z a£ 0 s i n c e p (x,0,0) ^  T,, so ac + c, + b = 0 and b « c(ac+l) . x fl a b Thus XS^S^ i s a r e g u l a r i n v o l u t i o n i f X = I 0 c \0 1 where c = p o r p and b c ( a c + l ) . Case I has 6 d i s t i n c t X's. Case I I has 12 d i s t i n c t X's. Case I I I has 12 d i s t i n c t X's. Case 17 I) has 36 d i s t i n c t X's. IV I I ) has 24 d i s t i n c t X's. T o t a l 90 d i s t i n c t X's. P r o p o s i t i o n . There are 43,470 - 90 x 23 x 21 r e g u l a r i n v o l u t i o n s i n M 2^. Pr o o f . L e t R = { ( r ^ r ^ ) ( * y r 2 ) : r 2 € T 5 , r ^ r ^ i f i f c j , r 5 - q 5 , r ? - q 4 o r r ^ = q^ and r 4 = q^}. U = {a : a i s a r e g u l a r I n v o l u t i o n o f M 2^}. L e t ( q 5 , r 4 ) ( r 5 , r 2 ) e R. I f r ^ = q^ then t h e r e are 22 c h o i c e s f o r r ^ and f o r each o f these c h o i c e s t h e r e are 21 c h o i c e s f O r r 2« Hence there 55. are 22 x 21 d i s t i n c t elements of R o f the form (35^4) ( < I ip r 2)' I f r 3 ^ q 4 1 t h e n r 3 = q 3 8 1 1 3 r 4 = q 4 * Tftere are 21' c h o i c e s f o r r 2 and thus 21 d i s t i n c t elements o f R o f the form (q^q^)(q.yT 2) . T h e r e f o r e |RJ = 22 x 21 + 21 = 23 x 21. F o r each ( r ^ r ^ ) ( r ^ , r 2 ) e R there e x i s t s X € M 2^ such t h a t r ^ — ^ q ^ ^ 1 = 2, 5 s i n c e i s 4 - f o l d t r a n s i t i v e on F o r each r e g u l a r i n v o l u t i o n € G-^ S-^ S,- k =» 1, 90 1 CTk g k l e t a k =» Xu^X . a k i s an i n v o l u t i o n and r ^ — - ^ - r ^ , r ^ - r ' >-r 2 < 1 Hence f o r each ( r ^ , r ^ ) ( r ^ , r 2 ) e R t h e r e are a t l e a s t 90 r e g u l a r i n v o l u t i o n s a such t h a t r ^ ^ ^ - ^ r ^ , r ^ < CT:>r2. I f T i s a r e g u l a r i n v o l u t i o n o f Mg^ and r ^ - ^ - I ^ - r ^ , r ^ - s v ^ > r 2 l e t u - X _ 1 T X . U i s a r e g u l a r i n v o l u t i o n and q^ «*-^ -5»«qi|., q^^r^-^qg. T h e r e f o r e p. e G-^S^S^ so u * u^. f o r some k = 1, 2, 90. Hence T = a k f o r some k. T h e r e f o r e f o r each ( r ^ , r ^ ) ( r ^ , r 2 ) e R there are e x a c t l y 90 r e g u l a r i n v o l u t i o n s cr such t h a t r^^sS^r^, r^.< 0>.r 2. L e t T e U. Then T i s a product o f 12 t r a n s p o s i t i o n s . r^_ = q^ appears i n one o f these t r a n s p o s i t i o n s . I f q,_—"L-^q^ then q^ must appear i n another one and T = ( q 5 , q l f ) ( q 3 , r 2 ) ( u 1 , u 2 ) ... ( U ^ ^ , U 2 Q ) , u j c * r 2 e ^2* I f q,=—>r^ 4= q^ then q^ must appear i n another t r a n s p o s i t i o n and T = ( q ^ r ^ ) ( q ^ , r 2 ) (u-^Ug) ... . ( u ' ^ u " ^ ) • T h e r e f o r e f o r each T e U there e x i s t s ( r ^ r ^ ) ( r ^ r 2 ) e R 56. such t h a t T - ( r ^ r ^ ) (ryr2) (^tUp) ... ( u ^ u ^ l Therefore | u | » 90 x |R| » 90 x 23 x 21 - 43470. 1 A B 5.10 Lemma. I f X » | 0 D E ) and D » 0 i f f :E + 0", ,0 E 2 D 2 Then X S ^ = S ^ X ( i . e . X € C ( S 5 S 5 ) ) . . Proof. 3 5 2 2 x+y z y z /x + Ay + Bz Dy + Ez \ E 2 y + D 2z / x + y 2 z 2 + Ay + Bz^ Dy + Ez \ E 2 y + D 2z / 'x+Ay+Bz+ [DE 2y 2+ED 2z 2+ ( E 5 + D 5 ) y z j 2 Dy + Ez 2 2 E y + D^z 2 2 ( 'x + Ay + Bz + y z Dy + Ez E 2y + D 2z ' since D=0 or E»0 hut not both. Hence S,SCX • XS,S,-. 3 5 3 5 5.11 P r o p o s i t i o n . I n p r o p o s i t i o n 5 . 8 , i ) A l l the 6 i n v o l u t i o n s of Case I are conjugate i i ) A l l the 12 i n v o l u t i o n s of Case I I and the 12 i n v o l u t i o n of Case I I I are conjugate. p i l l ) I n Case IV i ) , f o r each x«0, p , p , a l l the 12 p i n v o l u t i o n s w i t h b«ad+xd are conjugate. Iv) I n Case IV i i ) , f o r each x«0, p , p 2 , a l l the 8 i n v o l u t i o n s w i t h b«ac +xc are conjugate. 57. Proof, i ) L e t X-: ax Ix' X. _ „CT= ax a 2 x 0 1 a f .0, 2 x»0 or 0 , a 2 x 0 Q 2 •2 a x G _2 A 0 0 a = 0 a 0 \o 0 a Hence a X l x - X ^ a , so a X ^ S ^ - X ^ c r S . ^ - X ^ S ^ a , since a e C ( S ^ ) . .2N Therefore S 5 X 1 0 S 5 - S " 1 X i p S 5 - X l p 2 . So a l l 6 i n v o l u t i o n s of type I are conjugate. 1 d b\ / l a e 1 i i ) L e t X 0 1 d d ^ 0 and \x L e t a db adb X10 ae 0 1 0 ] e^. 0. vO e 1> 1 0 ae' ae 0 0 e l e C ( S 5 S 5 ) L0 0 1 bd^+d 0 d d' 0 d' 58. Xdb adb / l d+bd 2 bd 5+b^ o - - 2 d 0 1 bd 2+d d 0 0 Hence o ^ X - ^ S ^ » X ^ O ^ S ^ . X ^ S ^ a ^ . T h e r e f o r e a l l the 12 i n v o l u t i o n s o f type I I I are conjugate. T a e X 1 0 a 1 1 1 0 ae 0 1 1-0 0 e 0 0 1/ \o e 2 0 1 0 ae^+e .2 o / \o 0 U T = K a e ae 1 0 a e 2 \ 1 a e 0 0 1 0 0 e 20 / \o e 1 3 2 N a+ae^ e+ae 2 e e \ o : e 2 0 0 e+ae' e 2 e e 2 0 Hence T ^ X ^ S ^ - M a e ^ S ^ = M ^ V a e ' T h e r e f o r e a l l the i n v o l u t i o n s o f type I I a r e conjugate t o X 1 0S^S^. T h e r e f o r e a l l the 24 i n v o l u t i o n s o f type I I and I I I ar e conjugate. I i i ) L e t X X 0 1 x a a d : a a d X a d x a 0 ad ,2 „ d' 0 e c(s 3 s 5 ) ad +ad +xd' a2 o 59. 1 ad 2 x 0 0 d 2 ) since d ± 0 . ,0 d 0, Hence X 0 1 x a a d S 3 S , - = W ^ V a d ' a a d X a d x S 3 S 5 * p Therefore f o r each x•» 0 5 p or p , a l l the 12 i n v o l u t i o n s 2 of type I"? i ) , w i t h h ^ ad + xd are conjugate. 1 a ac 2+xc i v ) L e t \ a c x » l o c c I , a a / l 0 0 1 0 I e C ( S 5 S 5 ) 0 1 c / \ o 0 1/ 1 0 a \ / l 0 xc W a - 1° 1 0 0 C C 0 0 1 / \ 0 1 c cr X = a acx since 1+c 2 =» c. Hence V ^ S ^ = X ^ ^ S ^ = X ^ S ^ S ^ . 2 Therefore f o r each x » 0, 0 or 0 , a l l the 8 i n v o l u t i o n s of type IV i i ) w i t h b » ac +xc are conjugate. 5.12 P r o p o s i t i o n . I f a i s a r e g u l a r i n v o l u t i o n of Mg^ then a does not l i e i n the centre of any 2-Sylow subgroup of Mg^. Proof. The r e g u l a r i n v o l u t i o n s of Mg^ l i e i n m conjugate c l a s s e s K,, K 0. — , K . L d m 60.. L e t L4';-. K 4 '"n G, S,S,_ = [at a 6 K. and q^«< CT »»q,ij,r CT ^Qg 3' By. p r o p o s i t i o n ' 5.11, L^ must c o n t a i n a l l s i x or none o f the i n v o l u t i o n s o f Case I i n the p r o o f of p r o p o s i t i o n 5 . 8 . S i m i l a r l y , L^ c o n t a i n s a l l 24 or none o f the i n v o l u t i o n s o f Case II and Ills none, 12, 24, or 36 o f the i n v o l u t i o n s o f Case IV i ) ; and none, 8, 16, or 24 of the i n v o l u t i o n s o f Case I v i i ) . T h e r e f o r e | L j | a 6a + 24b + 12c + 8d where a » 0 or 1, b » 0 or 1, c a 0, 1, 2, or 3 and d = 0, 1, 2, or 3 . T h e r e f o r e |L^| i s even. As i n the proof of p r o p o s i t i o n 5 . 9 , l e t R:•- { ( r 5 , r 4 ) ( r 3 , r 2 ) : r ± e T 5 , r ± ^ i f ifj, " r ^ - a ^ , r 3 » q 4 or r^«q^ and r^=q^}. F o r each ( r ^ , r ^ ) ( r ^ , r g ) e R th e r e e x i s t s X s Mgij. such t h a t "r^- X > q ^ ; i • 2, 3 , 4, 5 . L e t CT e L j . Then XCTX" 1 € K j and X c r X - 1 i s an i n v o l u t i o n and XCTX" 1 a X T X " 1 I f f CT = T. Assume t h a t f o r some ( r ^ , r ^ ) ( r ^ , r 2 ) e R and (t,~,t^) ( t y t ^ ) 6 R and some X,u e M 2^ such t h a t r ^ — h ^ q ^ f t . ^_JL^.q. i a 2, 3 , 4, 5 and some CT, T e L . we get 7 - l -1 • *J XC T X - 1 a ^ T U " 1 . r ^ u g . X g X > r ^ and t,-«H T^ > t ^ and r ^ a t,_ * q^ by d e f i n i t i o n o f R. Hence r ^ a t ^ . Assume r ^ f t ^ . Then by d e f i n i t i o n o f R, i ) r ^ « q^ and t ^ a q^ and t ^ a q^ or i i ) r- 5 - q^ and r ^ a and - q^. But i ) i m p l i e s t h a t r ^ = t ^ c o n t r a r y t o r ^ a t ^ and i i ) i m p l i e s t h a t » c o n t r a r y t o r ^ a t ^ . XCTX -! T h e r e f o r e r , a t-.. Hence r 0 a t A because r-,*C ->r 0, , 3 3 2 2 3 2' t J£EHL»t 61. T h e r e f o r e ( r ^ r ^ ) ( r ^ , r 2 ) = ( t ^ t ^ ) ( t ^ t g ) . Hence d i s t i n c t elements of R \ g i v e r i s e t o d i s t i n c t i n v o l u t i o n s . T h e r e f o r e |K^| > JL^| . JR| f o r each . J • 1, 2, ' m, By p r o p o s i t i o n 5 . 9 , the number o f r e g u l a r i n v o l u t i o n s of M u = m m m * - 23 x 21 x 90 m E |K J > 2 |L J R m |R| E |L.| - 23 x 21 x 90 J - l 3 " j - 1 J j-1 J s i n c e t h e r e are 90 r e g u l a r i n v o l u t i o n s i n G-^ SyS^ . T h e r e f o r e JKjj » JL^j |R|. L e t | L j | - 2n^ (|L^| i s even). I f CT i s a r e g u l a r i n v o l u t i o n o f then a € f o r some j and the number of conjugates of CT » |K^| [ M , ^ . : C ( C T ) ] . | M 2 ^ | M 2 L F 2 1 0x3^ x 5 x 7 , x l l x 2 3 Hence |C(a) |Kj| |Lj||R|- 2n^x23x21 2 9 x 3 2 x 5 x l l L e t P be a 2-Sylow subgroup of M 2 l r The |p| » 2 1 0 . But 10 2 does not d i v i d e | C ( C T ) | . T h e r e f o r e P qr C(cr) so CT £ Z(P) . T h i s completes the p r o o f of theorem 5 . 2 . 5.13 P r o p o s i t i o n . There a r e no i n v o l u t i o n s o f f i x i n g e x a c t l y 2 l e t t e r s i n T c . 5 P r o o f . Assume CT i s . a n i n v o l u t i o n f i x i n g e x a c t l y 2 l e t t e r s r ^ and r,- s T^. L e t r 2 , r ^ e T,- such t h a t r g ^ - ^ r ^ . There e x i s t s \ e such t h a t r f e — q >q^ k«2, . 5 s i n c e M 2^ i s 4 - f o l d t r a n s i t i v e on T,-. L e t u « X - 1CTX. Then u i s an i n v o l u t i o n f i x i n g e x a c t l y q^ and q,_ and q 2 - . ^ >q.y T h e r e f o r e \s c G ^ c Gy (The p r o o f of t h i s i s i d e n t i c a l t o the p r oof of g € G ^ S ^ i n p r o p o s i t i o n 5 . 7 , l i n e s * t o / l a b \ L e t u =» gS^ where g » I 0 c d \0 e f / gS, € G-, 2 2 2 2 2 2 2 x + a y + b z + cey + d f z + y z 1 c 2 y 2 + d 2 z 2 e 2 y 2 + f 2 z 2 /x^ + (aN-ce)y + (b>df)z'f + yz' ' c 2y 2 + d 2z 2 e 2y 2 + f 2 z 2 U 2 + (a2+ce)¥2 + (b2+df ) ¥ 2 + W N c ^ 2 + d V e ^ 2 + f V s i n c e u =» gS^ i s an i n v o l u t i o n . Hence z.»eV + f V . e2(cy+dz) + f 2(ey+fz) « e ( e c + f 2 ) y + ( e 2 d + f 5 ) z y « c ¥ 4 d V » c 2(cy+dz) + d 2 ( e y + f z ) - ( c 5 + d 2 e ) y + d ( c 2 + d f ) z . These must h o l d f O r a l l v a l u e s o f y and z (not b o t h z e r o ) . I f y«0 and z«l then e 2 d + f 5 » 1 d ( c 2 + d f ) m 0 I f y = l and z»0 then e(ec+f ) =» 0 3 2 c +d e a 1 g £ G 1 so det g • c f + de M 1 (1) (2) (3) (5) 63. I f e=0 or (3=0 then, (5) i m p l i e s c f - 1 . Hence f » c 2 J> 0. I f efo and d^O then (2) i m p l i e s e 2+df » 0. Hence f =* c 2 d 2 . 2 A l s o (3) i m p l i e s ec+f » 0 so ec+cd » 0. Therefore e==d' oroa 2 c=0.n,"If ('' 6=s0 -then f=0 ••-•and (1) Implies e d = 1 so d=e. 2 But (5) i m p l i e s de = 1. Hence d -'-1 sO d• - 1 » e I f d»e then (1) i m p l i e s f«0. Then c= 0 and d • e - 1. Therefore f » c 2 and c f = 0 i f f d = e •> 1. x * U 2 + (a 2+ce)V 2 + (h 2+dc 2)W 2 + W .'- x + ( a + c 2 e 2 ) y + (b+d 2c)z + y 2 z 2 + (b 2+dc 2) (ey+c 2z) + (a 2+ce)(cy+dz) + c 2 e 2 y + d 2 c z + y 2 z 2 2 2 2 2 » x + (a•:•+ eh + c de + a c + c e)y + (h + e 2 ! ) 2 + cd + a 2 d + cde)z. Therefore & + eb + c de + a c + c e » (6) b + c 2 b 2 + cd + a 2 d + cde =» 0. (7) Case I . e => 0 and d = 0, Then c f 0 and 2 2 (6) i m p l i e s a + a c + 0 so a « 0 or a » c . 2 2 (7) Implies b + b c =» 0 so b =* 0 or b • c . P 2 2 2 2 x^ + a'y* + b^z* + yz ^ c z 2 I f a » 0 then ( c ) g S 5 ^ c 2 c 2 - ( c j so gS^ f i x e s V0 / \ o / (1, c, 0 ) , I f a f 0 then a = e 2 |= 0 i f b = 0 then 1 \ / 1 c 2 / 4> \ c [ e 2 ] 2 Hence gS^ f i x e s ( 1 , 0, c 2) / 2 2 2 p\ / 2 , fp + C C '+ C .C + C.C V-/P +1' C . C C.C 2 Hence . gS^ f i x e s (p, c, c ). I n each case u » gS^ f i x e s some (x, y, z) e T , contrary t o the c o n s t r u c t i o n of Case I I . e - 0 d =j= 0. Then c f 0 2 2 and (2) i m p l i e s c + dc = 0 Hence d =* 1 since c j . 0. 2 2 (6) i m p l i e s a + a c = 0 so a = 0 or a = c . x\ / x 2 + a 2 y 2 + ( b 2 + c 2 ) z 2 + yz I f a - 0, I f a - c 2 , I n both cases, gS^ f i x e s some (x, y, z) e T Q, contrary to the c o n s t r u c t i o n of n. Case I I I , d =» 0 and e ^ 0. Then c ± 0. and (3) Implies ec + c-.=» 0. Hence e - 1 since c j . 0 (7) i m p l i e s b + A 2 = 0 so b « 0 o r b = c. 65. / 2 , 2 v 2 T.2 2 > fx + (a +e)y + b z + yz ' .2 c y 2 2 y + cz i f b == 0, 0 0 '1 c c 0\ /0 +1> i f b » c = 1, I 0 — > 0 0 / 2 2 2 2 /x + (a +c)c + c + c i f b > c = 0 or 0 2, ( c I—>( c 2 . c 2 / 2 2 2 N fx + a c c 2 2 2 2 But (6) i m p l i e s a + c + a c + c » 0. Thus a + a c » 0; " « • 2 so a = 0 or a =» c 2 2 2 i f a a 0 l e t x - l. Then x + a c =* 1 =* x p p p ? P i f a 'as c l e t x a* 0. Then x + a c » 0 +1 » p ss x. Hence gS^. f i x e s (x, c, 1) . I n each case, gSj- f i x e s , some (x, y, z) e T .. Case Iv". d f O and e f 0. Then c =» 0 and d » e =« 1. 2 2 (6) i m p l i e s a + b = o so a « b 2 2 2 2 x + a y + az + yz y | gS^ | z 2 z/ - \ y 2 66 < i f a - 0, i f a - 1, i f a = 0 or fj , Therefore i n each case gS^ f i x e s some (x, y, z) e T Q, contrary t o the c o n s t r u c t i o n of . u.a gS^. Therefore the assumption that there i s an i n v o l u t i o n a f i x i n g e x a c t l y 2 p o i n t s i s f a l s e . 5.14 P r o p o s i t i o n . There are no i n v o l u t i o n s of M 2^ f i x i n g e x a c t l y 4 p o i n t s i n T,_. Proof. Assume cr i s an i n v o l u t i o n of M 2^ which f i x e s e x a c t l y 4 l e t t e r s r 2 , Ty r^, r,_ e T^ there e x i s t s X e X such th a t > q k k = 2 y , 5. since M 2^ i s 4- f o l d t r a n s i t i v e on T c. L e t fji a X~ a X. Then ia i s an i n v o l u t i o n and u f i x e s e x a c t l y q 2 , q^, so u € G^. / l a b\ / l a(l+c)+be b ^ f ^ d ' l e t u =* o c d . Then | j 2 = 0 c 2+de d(c+f) \0 e f / \0; e(c+f) 2 2 2 u - 1, so c + de'a f + de • 1, f 2+de Therefore c f , so c a f . 67. 2 / 2 Hence u = 0 c +de 1 a(l+c)+be b(l+c)+ad G 0 0 c 2+de 1 G 0^  0 1 0 ,0 G li Case I . I f a « 0 and e » 0. Then detu » c a l , SO C a 1. Hence u f i x e s (x, y, Q) f o r any x,y c o n t r a r y t o c o n s t r u c t i o n o f u. Case I I . I f a a o , e 4s 0. Then be a o , so b >' 0, and 2 2 2 c + de a l , so d a e (1+c) . 1 0 0 0 c e' 0 e c Hence u = [ e 2 ( l + c ) 2 and e S(l+o)] i i \ i e 2 c ( l + c ) + e 2 ( l + e ) 2 \(l+c) + e / e 2 ( l + c ) | Thus, u f i x e s some (x, y, z) € T Q , c o n t r a r y t o c o n s t r u c t i o n o f u. Case I I I . I f e a o and a f 0. Then c . • 1 so c a 1. Hence ad a o , so d = 0. Il a b\ / l \ fl + ab + ab\ Il T h e r e f o r e p. a 0 1 0 j and \0 0 1 b ' * b a b a, so V f i x e s ( 1 , b, a) e T Q , c o n t r a r y t o c o n s t r u c t i o n o f u. 68. 2 2 Case IV. e ={= 0 and a f 0 . Then d » e (1+c) . i ) i f c = 1 then d = 0 so be" - 0 . Hence b = 0 . 1 a 0 T h e r e f o r e u - | 0 1 0 J and ,0 e 1, i i ) i f e -• 0 then Then a + be - 0 A a T h e r e f o r e u - 0 0 * \o e o / i i i ) i f c :-' P o r 0 2 2 2N 1 + ae + ae 2 e e^ 2 2 - c, (1+c) - c . 2 2 2 2 Hence d =» e c. Then ac + b e 0 , so b =» ac e . (1 a a e 2 c 2 \ / l \ /l+a+a\ 0 c ,e 2c J arid I i )—li>( e+c 2 j 0 e c / \ec/ \e+ec 2/ T h e r e f o r e f o r each p o s s i b i l i t y , u f i x e s some (x, y, z) e T , c o n t r a r y t o c o n s t r u c t i o n o f u. T h e r e f o r e the assumption t h a t CT f i x e s e x a c t l y 4 l e t t e r s o f T c i s f a l s e . 5 5.15 P r o p o s i t i o n . I f CT i s an i r r e g u l a r i n v o l u t i o n o f M2ij. then CT f i x e s e x a c t l y 8 l e t t e r s and CT ~ u e H and a, b € GF(4) Mi a 0 o r b ± 0 , 6 9 . and there are 15 such p's. Proof. L e t CT he an I r r e g u l a r i n v o l u t i o n of Mg^* Then CT f i x e s at l e a s t 5 l e t t e r s r . , ..., r _ (by P r o p o s i t i o n 5.13 and 1 5 5.14). There e x i s t s X e M L e t u Hence |i e H such th a t r f c . -1 2 X cr X. Then u » . 1 and u f i x e s q 1 , q 1,. 5* L e t |i =» SO [X Hence u "1 0 a(l+b" 1)^ Therefore u 2 => ( 0 b 2 c(b+b" 1) ,0 0 -2 and , 5-Therefore (x, y, z) e T Q i s f i x e d by u i f f z « p. Therefore 2 u f i x e s e x a c t l y ( 1 , 0 ) , ( 1 , p , 0 ) , ( 1 , -1, 0) i n a d d i t i o n to q x , q 2 , qy q^, q,_. Hence u f i x e s e x a c t l y 8 l e t t e r s of T^ so CT f i x e s e i g h t . There are f o u r d i t i n c t choices each f o r a and c, so there are 16 d i s t i n c t u's. But the choice a = 0 and c = 0 gives (a = I which i s not an I n v o l u t i o n . Hence there are 15 d i s t i n c t i n v o l u t i o n s i n H. 5.16 P r o p o s i t i o n . A l l the i n v o l u t i o n s of H are-conjugate I n G 2 < ' l 0 a\ 0 l b ) f o r a j» 0 or b |= G and ,0 0 1/ Proof. L e t X ab CT v- = ab 70. / l c 0 b(a+bc) a(a+bc) 0 \0 0 a+bc^ where c = 0 i f a f.O and e = 1 i f " a » 0 . Then a+bc ^ 0 . , d e t C T a b = ( a +^ c) [a(a+bc) + bc(a+bc) ] » (a+bc) =« 1 so c r a b e Gg. / l 0 l \ / l c 0 \ (1 c 0 ' CTabX10 3 8 G 1 0 b(a+bc) a(a+bc) 0 ] = b() a() 0 \ 0 0 I/ \0 0 (a+bc)/ \ 0 0 () y 1 c o \ / 1 0 a \ /1 c a+bc X. ho f t h - 1 b() a() 0 0 1 b - b() a() ab()+ab()] . o o ( ) / \ o o 1/ \ o o () Therefore CT^X10 - X a bCT a b, so X1Q » o£ X^o^. Hence a l l the i n v o l u t i o n s of H are conjugate i n Gg. ' ,^  5.17 P r o p o s i t i o n . A l l the i r r e g u l a r i n v o l u t i o n s of are conjugate I n M 2 l r Proof. I f cr I s an I r r e g u l a r i n v o l u t i o n then cr i s conjugate to some u e H ( P r o p o s i t i o n 5 . 1 5 ) , and a l l the. i n v o l u t i o n s of H are conjugate ( P r o p o s i t i o n 5.1.6) . Therefore a l l the i r r e g u l a r i n v o l u t i o n s of Mg^ are.conjugate. 5.18 P r o p o s i t i o n . I f A = [ t ^ s t g , t y t ^ , t^} c T^ then there are e x a c t l y 15 i n v o l u t i o n s i n Mg^ each of which f i x e s each element of A. Each of these i n v o l u t i o n s f i x e s each element of some set B = { t x , ty V 1, Vg, v^}. 71 . Proof. There e x i s t s X e M 2^ such that t ^ — ^ q ^ 1=1, . . . , 5 because M 2^ i s 5 - f o l d t r a n s i t i v e on T^. L e t u^, ..., be the i n v o l u t i o n of H. L e t cr^ '.=• Xu^X.-1 k«l, ..., 15 . Then f i x e s A f o r each k»l, ..., 15 and each a f e i s an i n v o l u t i o n . I f a f i x e s A then u - X'^CTX f i x e s '"q^ , , q,_. Hence p = u f c f o r some k, so a a f c f o r some k. L e t v^ be such th a t v ^ — ^ . P ' I i = l , 2, 3 , where P l - ( 1 , .1, 0) p 2 = ( 1 , p, 0) p 5.« ( 1 , p 2 , 0 ) . Each p^ i s f i x e d by f o r each k. Therefore f i x e s v i 1=1, 2, 3 , f o r each k. So a k f i x e s { t ^ , tp., v-j., v 2 , v^} f o r each k = l , 15 . 5.19 D e f i n i t i o n . A S t e i n e r system <^(<t, m, n) i s a set of m-member clubs formed form n I n d i v i d u a l s subject t o the p r o v i s o t h a t eve;ry l persons must meet together i n one and onlycone,club. The S t e i n e r Group of (£ ( l t m, n) I s the group of a l l permutations on n i n d i v i d u a l s which leaves the set of clubs of m, n) i n v a r i a n t . 5.20 Remark. W i t t [7] proves t h a t M 2^ i s the S t e i n e r group f o r <^(5, 8 , 2k) t Stanton [6] s t a t e s t h a t and M 2 2 are the S t e i n e r groups f o r <=^(4, 7, 23) and ( 3 , 6, 22) respec-t i v e l y .and .that <E>(<t, n) has C(")/C(^) elements ( c l u b s ) . We make use of these r e s u l t s i n the f o l l o w i n g but make no attempt to prove them. 5.21 P r o p o s i t i o n . There are 11,385 » 3 2 x 5 x 11 x 23 i r r e g u l a r i n v o l u t i o n s i n M2^. 7 2 . P r o o f . L e t A » [t^, t g , t^> t ^ , t^} c T^. There i s e x a c t l y one B ec=?(5, 8 , 24) such t h a t A c B. L e t B = (t.^, — , tg}. By p r o p o s i t i o n 5.18 there are 15 i n v o l u t i o n s 0 ^ , a 2 , ..., a-j^, each o f which f i x e s each element o f A. L e t C f e he the Image of B under a f c. Hence A c f o r each k. A l s o G k € <E?(5> 8 , 24) s i n c e elements o f Mg^ map c l u b s o f <^(5, 8 , 24) onto o t h e r c l u b s . (M 24 i s the S t e i n e r group of < » ( 5 , 8 , 2 4 ) ) . Hence C k - B f o r each k - 1 , 2, ..., 15 because by d e f i n i t i o n o f CS? (5, 8 , 24) the r e i s e x a c t l y one club o f <~>{5, 8, 24) c o n t a i n i n g A. Assume t h a t f o r some n, 1 < n < 15, cr does not f i x — — • ' n each element o f B. Then a n i n t e r c h a n g e s two o f them, say a t j and t g (and f i x e s the ot h e r s ) i . e . t ^ < >-tg. But an f i x e s 8 l e t t e r s so i t must f i x some u^, Ug £ B. Hence each a f c f i x e s u 7 and ug because by the pr o o f o f p r o p o s i t i o n 5 .18, each a f e f i x e s the same 8 l e t t e r s . T h e r e f o r e t ^ - * ^»tg f o r each k = l , 2, ...,15 because each a k maps B onto i t s e l f . Hence cr^ a 2 f i x e s and t g . As i n the pr o o f o f p r o p o s i t i o n 5 .18, there e x i s t s X € Mg^ such t h a t X " 1 a k \ • [x^_ k=l, 2, 15 where the u^'s are the i n v o l u t i o n s o f H. {u-j., , u ^ , 1} i s a grOup so {a-j_, ..., a-^y 1} i s a l s o a group. T h e r e f o r e 0 ^ a 2 » a m f o r some m, 5 £ m <_ 15. Hence a m f i x e s t ^ and t g c o n t r a r y t o t 7 < -ytg f o r a l l k=l, 15 . Hence the assumption • a t h a t t y < n > t g f o r some n i s f a l s e . T h e r e f o r e each cr k f i x e s each element o f B. 73. Conversely, f o r each B e <^(5, 8 , 24) there are exactly 15 involutions, each of which f i x e s each l e t t e r of B. (Choose some A =* {t^-, t,_} c B and construct the f i f t e e n o"k's as i n the proof of proposition 5.18) . Also i f B.^ Bg € ^ ( 5 , 8, 24) and B^ ={= Bg then there i s no in v o l u t i o n f i x i n g "both B^ and Bg because such an i n v o l u t i o n would have to f i x at l e a s t 9 l e t t e r s / o f T^. But each i r r e g u l a r i n v o l u t i o n f i x e s exactly 8 l e t t e r s . Therefore the number of i r r e g u l a r involutions i n M24 ™ x t n e number of clubs i n G ? ( 5 , 8 , 24) » 15 x x 5 1 x ? : m 3 2 x 5 x 11 x 23 - 11 ,38$. 191X5! 8 ! 5.22 Proposition. I f CT i s an irregular. Involution of Mg^ then | C (C T ) | = 21,504 =» 2 1 0 x 3 x 7 and CT e Z(P) f o r sOme 2-Sylow subgroup P of Mgj^. Proof. Let CT be an Irregular i n v o l u t i o n of Mg^ 2 3 x 5 x 11 x 23 » the number of i r r e g u l a r Involution of Mg^ i » the number of conjugates i n Mg^ of CT - [M 2 4 : C (C T ) ] . Hence | C(CT) | ; « J ^ - , 2 1 0 x 3 ? x 5 x 7 x l l x 2 3 _ j>U x 5 . x 7 :. 3 x5x11x23 3 x 5 x l l x 2 3 Let P be a 2-Sylow subgroup of C(CT). Then |P| - 2 1 0 , and P i s a 2-Sylow subgroup of Mg^. Hence CT e x~ 1 ; Px: f o r some x e C(CT) since a l l the 2-Sylow subgroups of C(CT) are conjugate i n C(CT) and CT must l i e i n at l e a s t one of them. But XCTX 1 = CT f o r x e C(CT). 7*. Therefore a « xax e P, so a e Z(P). Th i s completes the proof of Theorem 5.4. 5.23 P r o p o s i t i o n . A i l the i n v o l u t i o n s of Mg-^  are conjugate i n Mg^. Proof. L e t a he an i n v o l u t i o n of M0-,. Then a i s an G — — , . 23 i n v o l u t i o n o f Mg^ aha a f i x e s q,-. Hence cr f i x e s \ 7t l e t t e r s £t^,' t^.} c T^. There e x i s t s X € Mg^ such th a t t ^ . — " " ^ ^ i 1=1, 4 (and q ^ — ^ q ^ ) > sin c e Mg-^  i s 4 - f o l d t r a n s i t i v e on T^. L e t ^ » X CTX, \I f i x e s {q^, q^}. Then u i s an i n v o l u t i o n o f H. A l l the i n v o l u t i o n s of H are conjugate i n Gg c Mg^, so a l l the i n v o l u t i o n s , of Mg^ are conjugate i n Mg^. 5.24 P r o p o s i t i o n . A i l the I n v o l u t i o n s of Mgg are conjugate I n Mgg. Proof. The proof i s f o r m a l l y the same as the prc^of :of p r o p o s i t i o n 5 . 2 3 . Replace 23 hy 22, 24 hy 23, 4 hy 3 , and 7 hy 6 . 5.25 P r o p o s i t i o n . There are 3795 - 3 x 5 x 11 x 23 i n v o l u t i o n s I n Mg^ and 1,155 » 3 x 5 x 7 x I I i n v o l u t i o n s i n Mgg. Proof. An arguement f o r m a l l y ' i d e n t i c a l t o P r o p o s i t i o n 5 .21 shows t h a t the number o f i r r e g u l a r i n v o l u t i o n s o f Mg^ » 15 x the number of Clubs i n <^(4^ 7 , 23) » 15 x x ! H l 2 L a 3 x 5 x 11 x 23 19!x4j 7! The number of i r r e g u l a r I n v o l u t i o n s , o f Mgg a 15 x the number of clubs i n <2?(3, 6, 22) -75. - 15 x — x - 3 x 5 x 7 x i l . 1 9 ! x 3 ! 6» 5.26 P r o p o s i t i o n . I f CT i s an i n v o l u t i o n o f M 2 j then CT e Z(P) f o r some 2-Sylow subgroup P o f and |C~(CT) | = 2,688 a 2 7 x 3 x 7 . • P r o o f . As i n P r o p o s i t i o n , 5 .22 [C-(CT)1 a ' M ^ ' a 2 7 X 3 2 X 5 X 7 X 1 1 X 2 5 , 2 7 x 3 x 7 , 3x5x11x23 3x5x11x23 L e t P be a 2-Sylow subgroup o f C~(cr) . Then |P| a 2 7 , so P i s a 2-Sylow subgroup o f M 2^ and CT e Z ( P ) . 5.27 P r o p o s i t i o n . I f CT i s an i n v o l u t i o n o f M 2 2 then |C5(cr) | a a 384 a 2^x3 and cr e Z(P) f o r sOme 2-Sylow subgroup P o f M 2 2 < P r o o f . As i n P r o p o s i t i o n 5-22, 1gCa)I - ' M 2 2 ' - 2 7X3 2X5X7X11 , g 7 x 3 . 3 x 5 x 7 x l l 3x5x7x11 L e t P be a 2-Sylow subgroup o f C(cr) . Then |P| - 2 7 , so P i s a 2-Sylow subgroup o f M 2 2 and cr e Z(P) . 76. B I B L I O G R A P H Y 1. Charmiehael, Robert D. I n t r o d u c t i o n t o the Theory of .Groups of F i n i t e Order. Hew York, Dover,, 195b j. c opyright 1937 J. 2. Coxeter, H.S.M. and Moser, W;O.J.-.. Generators and R e l a t i o n s  f o r E l s c r e t e Group s. B e r l i n , S p r inger-Verlag, 1957. 3 . Dickson, Leonard Eugene. L i n e a r Groups. Hew YOrk* Dover* 1958 [copyright 1 9 0 0 ] . 4 . H a l l , M a r s h a l l J r . The Theory of Groups. New York, Macmillan, 1959. 5. Mathieu, Emile. "Memoire sur l ' e t u d i e des Fonctions de P l u s i e u r s Q u a n t i t i e s . " J o u r n a l de Mathematiques Pures et  AppliquSes, 1861, pp. 241-323. 6. Stanton, R.G. 11 The Mathieu Groups." Canadian J o u r n a l of  Mathematics, v o l . 3 (1951), pp. 164-17^7" 7. W i t t , E r n s t . "Die 5-fach T r a n s i t ! v e n Gruppen von Mathieu." Abhandlung aus dem Mathematischen Seminar der Hansischen u n i v e r s i t y , v o l . .12 (1938). pp. 25b -2b4, : :  8 . Wong, W.J. "A C h a r a c t e r i z a t i o n of the Mathieu Group M, 0." Mathematische Z e i t s c h r i f t , v o l . 84 (1964), pp. 3 7 8 - 3 8 8 ^ Appendix A d d i t i o n t a b l e f o r G F ( 9 ) • a i s a p r i m i t i v e element o f GP(9) 2 8 a +a . - 1 a '. • 1 x+x+x 0 f o r each X s GF(9) + 0 a 2 a a.5 4 a a 5 6 a «7 cx 1 0 0 a a 2 a 5 4 a a 5 6 a «7 a 1 a ct a 5 1 4 a 6 a 0 • ct 2 a a 7 a 2 2 a 1 «6 a a? „7 ct 0 4 a .«* a? a3 . 4 a a V 2 a 6 a 1 0 a 5 1 a , 4 a 6 a 2 a 1 3 a •a7 a 0 a 5 0 V 6 a a 5 a 4 a 1 2 a a 6 6 a 0 1 a 7 4 a a 2 a 5 a a 7 7 a . 2 a '.4 a 0 a 1 a 5 a 3 ct 6 1 1 a a 5 0 ct 2 a a 6 4 a A d d i t i o n t a b l e f o r „GF(4). 0 i s a . p r i m i t i v e element of GF(4) 3B ^ 0 5 m 1 = G f o r each x € GF(4) + G 0 0 2 1 0 0 0 e2 . i 0 0 0 1 0^ * 0 2 p e I 0 0 , 1 i e2 0 0 

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