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Existence of algebras of symmetry-classes of tensors with respect to translation-in-variant pairs Hillel, Joel S. 1968

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EXISTENCE OP ALGEBRAS OF SYMMETRY-CLASSES OF TENSORS WITH RESPECT TO TRANSLATION-INVARIANT PAIRS by JOEL S. HILLEL B . S c , M . S c , M c G i l l U n i v e r s i t y , 1964 , 1 9 6 5 . A THESIS SUBMITTED IN PARTIAL FULFILMENT OF NTHE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f 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 s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1968 . 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 of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I a g r e e t h a t 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 r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t 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 of t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department or 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 or 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 not 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 . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada S u p e r v i s o r : R. Westwick. ABSTRACT The n o t i o n o f the ' c l a s s i c a l ' m u l t i l i n e a r maps such as the symmetric and skew-symmetric maps, has the f o l l o w i n g g e n e r a l i z a t i o n : g i v e n a v e c t o r - s p a c e V and a p a i r ( H ^ ^ ) where i s a subgroup of the symmetric group S n and ^ i s a c h a r a c t e r of , we c o n s i d e r m u l t i l i n e a r maps from V n ( n - f o l d c a r t e s i a n product of V ) i n t o any oth e r v e c t o r space, which are 'symmetric w i t h r e s p e c t to (^n^-^n)'^ i-e«* which have a c e r t a i n symmetry i n t h e i r v a l u e s on permuted t u p l e s of v e c t o r s , -where the permutations are i n B^ . Given a p a i r (H n,X n) and a ve c t o r - s p a c e V , ' (n) we can c o n s t r u c t a space VK ' over V through which the maps 'symmetric w i t h r e s p e c t to ( E f t * 3 ^ ) 1 l i n e a r i z e . The space V^ n^ i s u s u a l l y d e f i n e d a b s t r a c t l y by means of a c e r t a i n u n i v e r s a l mapping p r o p e r t y and g i v e s the te n s o r , symmetric and Grassmarm ..spaces f o r the ' c l a s s i c a l ' maps. Given a sequence of p a i r s U ^ ^ ^ ) } > 1 and the cor r e s p o n d i n g spaces V^ n^ , we l e t A = E V^ n^ (where n>_o yV°) i S ground f i e l d ) . In the c l a s s i c a l cases, A has a n a t u r a l m u l t i p l i c a t i v e o p e r a t i o n which makes A an a l g e b r a , i . e . , the Tensor, Symmetric and Grassmann a l g e b r a s . i i i . This presentation has been motivated by the attempt to generalize the construction of an algebra A to a wider family of 'admissible' sequences of pairs t (-^n*3^) } n>i * This consideration has l e d us to investigate permutation groups on the numbers 1,2,3,• which are closed under a c e r t a i n ' s h i f t ' of the fpermutations, i . e . , i f , a = (a-jj . .a g)(b^. . . b ^ ) . . . i s a permutation, we define = (a 1+l. . . a s + l ) Cb-^+1. . . b^+l).. . and we c a l l a permutation group H 'translation-invariant' i f for every a e H , a'"1-' i s also i n H . We begin our presentation by characterising the 'translation-invariant' groups. We show that the study of these ( i n f i n i t e ) groups can be reduced to' the study of c e r t a i n f i n i t e groups. Then, we proceed to discuss the l a t t i c e of the translation-invariant groups. F i n a l l y , we show that a translation-invariant group H , together, with an appropriate character X of H , represents an equivalence class of 'admissible' sequences of pairs U H ^ X ^ ) } n > 1 • For a p a r t i c u l a r choice of representatives ofi the equivalence c l a s s , we can construct an algebra of 'symmetry, classes of tensors' which generalizes the Tensor, Symmetric and Grassmann algebras. i v . TABLE OP CONTENTS Page CHAPTER 1 • 1 1. Translation-invariant groups 1 2. Closed subgroups and Generatives 8 CHAPTER 2 . 17 1. P r i n c i p a l and Minimal subgroups 17 2. P r i n c i p a l chains 22 3. Classes of generative elements 27 4. The length of a p r i n c i p a l chain 31 CHAPTER 3 41 1. Application: Admissible sequences 4 l 2. Strongly-admissible sequences 49 3. Algebras of Symmetry-classes of tensors 59 4. Basis of V[H,X] as a vector space 65 BIBLIOGRPAHY 70 V . ACKNOWLEDGEMENTS To Dr. R. Westwick, many thanks. CHAPTER 1 1. T r a n s l a t i o n - I n v a r i a n t groups. A problem i n m u l t i l i n e a r a l g e b r a concerning the c o n s t r u c t i o n of alg e b r a s of 'symmetry c l a s s e s o f t e n s o r s ' (see [ 6 ] ) , has le.d us to i n v e s t i g a t e permutation, groups which are ' t r a n s l a t i o n - i n v a r i a n t ' i n the f o l l o w i n g sense: L e t S m denote the group of a l l permutations of f i n i t e degree on the i n t e g e r s 1,2,3,... L e t a = (a, a 0...a„) ( b , b 0 . . . b . ) . . . be i n S . Then v 1 2 s v 1 2 t ' a> f o r an i n t e g e r i >_ o , we d e f i n e the ' t r a n s l a t e ' o f a to T i l be the permutation a - (a-^+i a^+i. .. a g + i ) (b^+i b^+i. . .b^+i). . . C l e a r l y , the d e f i n i t i o n of i s independent of the c y c l i c decomposition of cr , and V. i,j>_o , a , T e S we have ( a [ i ] ) [ J ] = a [ i+J] k n d ( a r ) 1 1 ] = a [ i ] T [ i ] . L e t H be a subgroup of S o (denoted by H < S^). Then, we d e f i n e f o r i >_ o , H ^ = {a-^-/a e H } . C l e a r l y , H'-1-' < S . 00 1.1 D e f i n i t i o n : L e t H < S . Then H i s s a i d to be t r a n s l a t i o n - i n v a r i a n t ( b r i e f l y , _H i s a t - i group) i f H ' " 1 - ' < H . I t f o l l o w s immediately from the d e f i n i t i o n that a n o n - t r i v i a l , t r a n s l a t i o n - i n v a r i a n t group has i n f i n i t e order. 2. [ i l I f H i s t r a n s l a t i o n - i n v a r i a n t , then so i s H , V i _> o . H i s s a i d to be reduced i f H i s e i t h e r the t r i v i a l group or i f H moves the i n t e g e r 1 . Unless the c o n t r a r y i s mentioned, by a t r a n s l a t i o n -i n v a r i a n t group we s h a l l always mean a reduced one. C l e a r l y , i f G i s a t - i group then, f o r some p >_ o , G = E^- where H i s a reduced t - i group. L e t H be a n o n - t r i v i a l t - i group, and l e t 0 be an o r b i t f o r the a c t i o n o f H on {l,2,3>--.} • Suppose 0 = {x 1,x 2,...} ,• then we l e t 0-^ = {x-^+d, x^+d, where r J i d >_ 6 , i . e . 0 J i s the corresponding o r b i t f o r the a c t i o n [ d] of H on {1,2,3,...} • Now, as H i s t - i , we have < H , hence, 0 ^ i s c o n t a i n e d i n one and on l y one of the o r b i t s f o r the a c t i o n of H . 1.2 P r o p o s i t i o n : L e t H be a n o n - t r i v i a l t - i group. Then the o r b i t s f o r the a c t i o n of H on {1,2,3,-..} are p r e c i s e l y Z 1,...,Z k , f o r some k >_ 1 , where Z^ = {i,i+k,i+2k,.. .} , 1 <_ 1 <_ k . Proof: V i > o , l e t 0^ denote the o r b i t c o n t a i n i n g i . Then, as H i s reduced, i t moves 1 and 0^ has more than one element. L e t k be the minimal d i f f e r e n c e between the i n t e g e r s i n 0 1 . Then, 0^*- 0 0 1 ^ 0 and so o| k - ' c o 1 . But then o j n k - ' c 0 V n>l , and so 1+nk e 0 1 V n>l , and by the c h o i c e o f k , the r e are no other i n t e g e r s i n 0^ . Now, i f k = 1 , then 0^ i s the on l y o r b i t , i . e . , H i s t r a n s i t i v e . We suppose k > 1 , and c o n s i d e r 0^ , 1 < i <_ k . As O ^ 1 " 1 ^ n 0 i ^ 0 , we have o j 1 " 1 ^ c 0 ± , hence, i+nk e G\ V n>l . L e t k^ be the minimal d i f f e r e n c e between the i n t e g e r s i n C\ ; then ^ <_ k . But, as [ k + l - i ] n Q ^ 0 W e have o f k + 1 " a - ' <= 0-, , and t h e r e f o r e l 1 ' l 1 ' k-j_ can not be l e s s than k , by m i n i m a l i t y of k 3 i . e . , 0^ =• { i , i+k, i+2k,. . .} V l£i<k . Q .E.D. When the o r b i t s f o r the a c t i o n of H are Z-^,...,Zk we say t h a t H i s of type-k. The t r i v i a l group i s s a i d to be of type-o. We can now proceed t o show the most important p r o p e r t y of t - i groups, namely, t h a t any t - i group must c o n t a i n a l l even permutations on each of the o r b i t s Z-^,...,Z^, We r e q u i r e some p r e l i m i n a r y remarks and d e f i n i t i o n s . L e t H be a t - i group. Then, the sequence of groups C H n } n > 1 , where V n>l , H n = {a € H / a(m) = m Y m>n} i s c a l l e d the a s s o c i a t e d sequence of H . A l s o , H = U H n>l n and V n>l , < H n + 1 and < H n + 1 . Suppose H i s of type-k, where k >_ 1 , and l e t ^ n ^ n > l l o e a s s o c i a ' t e d sequence. L e t k n denote the number of o r b i t s f o r the a c t i o n of on {1,2,...,n) V n>l . Then, i f i s the f i r s t n o n - t r i v i a l group of the sequence, we have k N >_ ^+-_ k n 2L • • * • 4. So, there e x i s t s M > N such t h a t k = k , f o r — n n+l every n >_ M . As H i s of type-k , i t f o l l o w s that k n = k f o r every n >_ M and so the o r b i t s f o r the a c t i o n o f on { 1 , 2 , . . . , n } must be Z± N> • • - >\ n where Z± N = Z± fV { 1 , 2 , . . . ,n} f o r 1 . <_ i <_ n and n >_ M . Let n - denote the group obtained by r e s t r i c t i n g the a c t i o n of PL to the o r b i t Z. . I t i s well-known n i , n t h a t H. „ a c t s t r a n s i t i v e l y on Z. _ and H i s a sub-i , n — i , n n d i r e c t product of n x . . . x H k n , f o r each n >_ M . (See [ 9 ] , 5 . 5 ) . Now, as < then < H . j n + 1 . A l s o , a s ^ < V l t h e n H t l 3 n < H i + l , n + l f o r 1 < 1 < k a n d  H k , n < ^ n + l ' The o r b i t s of H are Z-^,...,Z^. , where Z. = { i , i+k, i + 2 k , . . .} , 1 <_ i _< k . D e f i n e S. ( r e s p . , A^ ro ) to be the group of a l l permutations ( r e s p . , even permutations) of f i n i t e degree on Z^ , 1 <_ i _< k . Let S (k) = S n x...xS, and A (k) = A n x...xA, 0 0 l j » K,oo cox I j 0 0 k, In p a r t i c u l a r , - ^ ( 1 ) = A^ and S w ( l ) = S^ . We extend the d e f i n i t i o n by l e t t i n g A (o) = 1 = S (o) . 1.3 Theorem: L e t H be a t r a n s l a t i o n - i n v a r i a n t group of type-k 5 then A a >(k) < H < S Q o(k) . Proof: The case k = o i s immediate from the d e f i n i t i o n s . Assume H i s of type-k , where k > 1 , and l e t {H } . -. be J C — n J n > l the a s s o c i a t e d sequence. We have seen t h a t there e x i s t s M>1 CO such that < H ]^ nx...xH k n V n>M , where tL n i s a t r a n s i t i v e group on the i n t e g e r s i n Z. = Z. fl {l,2,...,n} V l<i<fc • > L e t A. (S. ) denote the a l t e r n a t i n g (symmetric) x,n v l j i i group on the l e t t e r s of Z. , V n>M , 1 £ i <_ k . We s h a l l prove t h a t , e v e n t u a l l y , A 1 n X . . . x A ^ n < H n < s l j n x - - - x S k j V n ; i . e . , e v e n t u a l l y , every c o n t a i n s a l l the even permutations on each of the o r b i t s f o r i t s a c t i o n . As H = U H , the theorem w i l l c l e a r l y be proved. n>l n F i r s t , we s h a l l prove that e v e n t u a l l y every component group H. must c o n t a i n A. . T h i s w i l l f o l l o w from the x y xi x j n well-known r e s u l t (see [ 2 ] , Chap. 10, Theorem 4.) t h a t i f an S-ply t r a n s i t i v e group i s of degree d , where S > 1 / 3 d + 1 , then such a group must c o n t a i n the a l t e r n a t i n g group. Then we s h a l l show that i n f a c t the groups A. ^ are a l r e a d y i n x ,n Choose m such that 1 + m k > M , and l e t o o — n t = 1 + (m + t ) k , f o r t = o , l , 2 , . . . . L e t G t = H l n V t>o ; then G^ i s a t r a n s i t i v e group on {1, 1+k,. . ., l+(m o+t)k} , i . e . , G^ i s a t r a n s i t i v e group of degree . d t = ^ mo + ^ + x ) v "^iP • Assume G^ i s , i n f a c t , S^-ply t r a n s i t i v e . Then we show the sequence of i n t e g e r s £ st^t>o i s s t r i c t l y i n c r e a s i n g . For any t v> o , H < H and n t n t + l so G^ . < • I n f a c t , Gfc . i s a subgroup of those permutations i n Gt_,-|_ which l e a v e the l e t t e r n^ +-_ f i x e d , i . e . , G t < t a e G t + 1 / a ( n t + 1 ) = n t + 1 ) . Then, S t + 1 must 6. be at l e a s t S t+1 (see [ 8 ] , Theorem 1 . 9 ) , i . e . , st+l — S t + 1 t — o V t>o and t h e r e f o r e , S+. > S^+t Y t>o . We have V t>o , G^ as an S^-ply t r a n s i t i v e group, where _> S Q + t , and of degree d t = (m Q + t + l ) . C l e a r l y , f o r s u f f i c i e n t l y l a r g e t , say t = , the i n e q u a l i t y S t > 1/3 d t + 1 w i l l h o l d V t>t± . Then, Y t M ^ , G t w i l l c o n t a i n the a l t e r n a t i n g group on i t s permuted l e t t e r s , -••*-> A i ; n t < h,nt Y ^ 1 ' B u t ' a s H l , n t < Yn^-i Y i>o , we have A, „ < HL V n>n, — x,n -L,n 2-By i d e n t i c a l argument we have t h a t A. < H _L $ i l J. j IX Y h>_n^ . , f o r some i n t e g e r n^ , 1 <_ i <_ k . "1 i Let n* = max(n, ,...,n ) $ then A. < H. k Y n>n* , 1 < i < k We have shown t h a t V n>n* , ^ < H i n x - • • x H k n and A. < H. < S. , 1 < i < k . I t remains to be x,n i , n i , n * — — shown that the A i n ' s a r e a c t u a l l y subgroups o f . Choose a prime p ^ 2 such that 1 + ( p - l ) k _> n* . Le t m = 1 + (p-l)k , hence Z 1 = [1,1+k,...,l+(p-l)k} and Z ± m = {i,i+k,...,i+(p-2)k} f o r 1 < i < k ; i . e . , Z-, i s an o r b i t o f l e n g t h p , wh i l e the remaining o r b i t s l,m - to ^ ' are of l e n g t h ( p - l ) . L e t (a-^ a 2 . . . a ) be any c y c l e of l e n g t h p , where a. e Z n . Such a c y c l e i s an even permutation, 3_ JL ^  in hence c o n t a i n e d i n . \ m . As i s the s u b d i r e c t product 7 -of Hn x...xH, m , i t f o l l o w s that there e x i s t s a e H , x , m K , m m such that a = a-.... a, where a. e H. and a, = (a.,... a ) 1 k i i,m 1 v 1 p' But as the orders o f a^,...,o^ are d i v i s o r s o f ( p - l ) ! , they are r e l a t i v e l y primed to p . Then f o r some r > o , a r = , i . e . , and consequently a 1 are i n . Hence, H^ c o n t a i n s a l l c y c l i c permutations of l e n g t h p on the i n t e g e r s i n Z, , _L <f III and as these c y c l e s generate A, , i t f o l l o w s that A l , m < Hm • Furthermore, as H^ < H m + 1 and H ^ < H m + 1 , then A± m + ± and A g m + 1 are i n H m + 1 . Proceeding i n d u c t i v e l y , we can show that f o r s u f f i c i e n t l y l a r g e n ( i n f a c t , n > m+k) , A, „,A 0 .. . . ,A, „ are i n H , i . e . event-— x,n c., n K j i i n u a l l y A ^ n x . . . x A ^ n < 1^ < S ^ x . . . x S ^ V n . As H = U H , then we have A (k) < H < S (k) . Q . E . D . n>l n With the a i d of the above theorem, we can g i v e a more 'symmetric' d e f i n i t i o n of a t - i group, i . e . , lA C o r o l l a r y : H i s t - i i f and on l y i f H ^ = H n s'-1-' —" • 1 1 1 - - . - C O Proof: The above c o n d i t i o n i s c l e a r l y s u f f i c i e n t . Suppose t h a t e H , f o r some a e S^ . We want to show that a e H . Assume H i s of type-k , k >_ 1 Then the o r b i t s f o r the a c t i o n of H are = {i,i+k,i+2k,. 1 < i < k . Hence, may be w r i t t e n as T i T 2 - , " T k where T . i s a permutation on Z. , 1 < i < k . Then, x x — — J 8. a = OjOg...o^ , where i s a permutation of , and c r f 1 ^ = T . l < i < k and aj-1- = T _ . 1 1 — K. JL Now, as a^1- e H , so does (a'- 1^)'• k" 1-' e H , i . e . , [k] [k] [k] Tk] „ „ a • a = a, aA • • -cr, i s i n H . Furthermore, as a . i s 1 2 k ' I T k l T kl a permutation on Z^ , then c l e a r l y , so i s a£ and aj_ has the same p a r i t y (odd or even) as CK , 1 _< i <_ k . f k l T h e r e f o r e there e x i s t s p. e A. such that a. J • p. = a. l l , 0 3 i i i Y 1 <_ i <_ k . L e t p = P i p 2 ' ' ' p k * ^ n e n P G A«>(k) ' a n d by the theorem p '4 H . Tkl T kl Thus a = a -p where a and p are i n H , i . e . , a e H . T h e r e f o r e H ' - 1 - ' = H n eo Q.E.D. In c o n c l u d i n g t h i s s e c t i o n we note that S and A r o are the o n l y t r a n s i t i v e ( i . e . , t y p e - l ) t - i groups. 2. C l o s e d subgroups and g e n e r a t i v e elements. As the main r e s u l t of the l a s t s e c t i o n , we have shown that f o r any type-k t r a n s l a t i o n - i n v a r i a n t group H , A o o(k) < H < S w ( k ) . T h i s enables us to reduce the study of the group H to that of the group H/A £ o(k) which i s a f i n i t e group, as 1 <_ [H : A j k ) ] _< [S w (k) : A < j o(k)] <_ 2 k . We can r e a l i z e these f i n i t e groups as f o l l o w s : L e t P(k) , k >_ 1 , denote the power s e t on { l , 2 , „ . . , k } , whose - t y p i c a l elements w i l l be denoted by ct,p,... . Let a A p , where a and £ e P(k) , denote the symmetric d i f f e r e n c e of the sets a and f3 . Then, {P(k),A} i s an a b e l i a n group whose zero element i s the empty set 0 , and such that a A a = 0 V a e P(k) . The group {P(k),A} w i l l be i n d i c a t e d simply as P(k) . L e t a = [a^,...,a^} e P(k) j then d e f i n e the • t r a n s l a t e ' of a , o r 1 , i >_ o , to be the set {(a^ + i )mod. k, . . . , (a^. + i)mod k} . L e t t i n g 0^^ = 0 , i t f o l l o w s t h a t a'-1-' e P(k) V a e P(k) and V i>^o . We extend the d e f i n i t i o n of a ^ by d e f i n i n g T -i 1 Tk i 1 a L J = a L J f o r i _> o . Then, V a,p e P(k) , and f o r a l l i n t e g e r s i , j , we have: 1. ( a [ l ] ) C j " ] = a C i + j ] 2. (a A = a C i ] A ( 3 [ i ] 3. a C k ] = a [ o ] = a . Let C be a subgroup o f P(k) , and l e t = [a-1-'/ a e C *} . C l e a r l y , C ^ < P(k) V i . 1 . 5 D e f i n i t i o n : A subgroup C .< P(k) i s s a i d to be t r a n s l a t i o n - i n v a r i a n t mod k i f c'-1-' < C . I f C i s t r a n s l a t i o n - i n v a r i a n t mod k , we w i l l say, 1 0 . f o r b r e v i t y , t h a t C i s c l o s e d . C l e a r l y , C i s c l o s e d i f and o n l y i f C = . The whole group C = P(k) and the t r i v i a l group C = 0 are o b v i o u s l y c l o s e d . Furthermore, the c o l l e c t i o n of a l l the c l o s e d subgroups of P(k) forms a l a t t i c e w.r.t. the o p e r a t i o n s fl , L) i . e . , i f C^ and are c l o s e d , then so are n C^ and U C^ (the s m a l l e s t subgroup o f P(k) c o n t a i n i n g C-^  and ). 1.6 Theorem: V kXL , there e x i s t s a n a t u r a l 1 - 1 c o r r e s -pondence between the type-k t - i groups and the c l o s e d sub-groups of P(k) . Proof: L e t k >_ 1 be given. Consider any a e S ^ k ) j then a can be w r i t t e n as °ja2'•-ak w h e r e a-}_ i s a permutation on Z^ = {i,i+k,i+ 2 k ,...} f o r 1 <_ i _< k . Now suppose t h a t among the a^,a^,...,a k , p r e c i s e l y a • ,a ,...,a are odd permutations. D e f i n e ^(a) = a l a 2 a t {a-j^a ,. . . ,a t} ; then 3 i s a map : S o o(k) -» P(k) . A l s o , i f a,T e S o o(k) , then i t f o l l o w s immediately from the d e f i n i t i o n of 3 , t h a t .3(CJT) = 3(a) A 3(T) ; i . e . , 3 i s i n f a c t a homomorphism o f groups. Now Ker .3 = [o e (k )'./Sr (a) =0}- ; i . e . , a e Sjk) i s i n Ker 3 , i f f , 0=0-^ a^. . .a^ where e v e r y CK i s an even permutation; i . e . , i f f a e A (k) . 11. Ther e f o r e , we have the u s u a l 1-1 correspondence between subgroups o f S (k) which c o n t a i n s A (k) and 00 CO the subgroups of P(k) . I t remains to be shown that the map 3 takes the type-k t - i groups onto the c l o s e d sub-groups o f P(k) . Now, 3 a s a t i s f i e s the pr o p e r t y : 3(a^ 1-') = (3 !(a))^ 1-', . V a € SA . Suppose t h a t H i s t - i group of type-k, i . e . , H ' ' 1 - ' < H and A o o(k) < H < S ^ k ) . Then, 3 (H ' - 1 - ' ) = (3(H) ) ^ < 3 ( H ) , hence 3 ( H ) i s a c l o s e d subgroup o f P(k) and c o n v e r s l y . Q.E.D. The homomorphism 3 d e f i n e d above i s c a l l e d the r e p r e s e n t a t i o n map. We can now prove f u r t h e r p r o p e r t i e s o f a t - i group H by determining the s t r u c t u r e of the c o r r e s -ponding f i n i t e , c l o s e d group 3 ( H ) . In p a r t i c u l a r , we show t h a t a t - i group i s completely determined by a unique permutation. L e t k >_ 1 be f i x e d , and l e t a e P(k) . De f i n e G(a) to be the c l o s e d subgroup generated by a ; i . e . , G(a) =P,n{e / 0 i s c l o s e d and a e C} . I t f o l l o w s t h a t G(a) = [a'- 1!"' A. . . Aa^tV i t i n t e g e r s , t = 1,2,...} , f o r the l a t t e r s e t i s c l e a r l y a subset of G(a) , and i s e a s i l y shown to be a c l o s e d subgroup. In p a r t i c u l a r , l e t t i n g 1 be the set {1} e P(k) , we have G ( l ) = P(k) and G(0) = 0 . G may be c o n s i d e r e d as a f u n c t i o n : P(k) - ( c l o s e d subgroup o f (P(k),A)} , where 12. a - G(a) . Le t a / 0 be i n P(k) ; then we denote by ||a|| the g r e a t e s t i n t e g e r i n the set a . We extend the d e f i n -i t i o n to a = 0 by l e t t i n g ||0|| = k + 1 . Then, 0 < ||a|| .<_ k + 1 V a e P(k) . Now, gi v e n a c l o s e d sub-group C , we d e f i n e M(C) to be an' element i n C w i t h the pr o p e r t y that ||M(C)|| ± \\a\\ V a e C . C l e a r l y , i f M(C) ^ 0 , then 1 e M(C) . Claim: . M(C) i s unique. For, i f ||M(C)|| = k + 1 , then M(Q) = 0 which i s unique. Hence, assume f|M(C)|| _< k and suppose t h a t f o r some a e C , |[a|| = ||M(C)|| . L e t p = a A M(C) j then, u n l e s s 0 = 0 , we have < ||M(C)|| which i s i m p o s s i b l e ; i . e . , p = a A M ( C ) = 0 , hence a = M(C) . Now we have then: M P(k) < s {Closed subgroups of (P(k),A)} . . G 1.7 P r o p o s i t i o n : For any c l o s e d subgroup C i n P(k) , G(M(C)) = C . Proof: The a s s e r t i o n i s t r i v i a l , i f C = 0 . Assume C ^ 0 and l e t a Q = M(C) . Consider any a e C ; then we must show t h a t a e G ( a Q ) = G ( M ( C ) ) . Now, as a e C , then ||a || <_ ||a|| . I f 13. [|ao|| = ||a|| , then by uniqueness of a Q , we must have aQ = a and we are f i n i s h e d . Assume l ! a 0 l l < l l a l l > and l e t d-j_ = | | a | | - | | a o l l • Define, a-. = a £ d l ^ A a . Then e C and Ha-JI < | | a | | . Now, i f | | a o H = Ha-JI , then a = a n = a ^ l - ' A a , i . e . , a = a ^ d - ^ A a which i s i n o 1 o ' 3 . o o G ( a Q ) , as r e q u i r e d . I f l l a 0 l l < I I a J I > l e t d 2 = Haj-.ll - | | a o | | , then d ? < &1 . Def i n e a^ = ao^^ ^ a i > a n ^ proceed i n d u c t i v e l y to d e f i n e a R = a ^ d n ^ A an_± where d R = H a n _ J I - l l a 0 H > V n>l . Then c l e a r l y , d-j^  > d 2 >...>: d n > . . . , and t h e r e f o r e f o r some m _> 1 , d m / o and d m + 1 = o . But d m + 1 = o i m p l i e s l l a m l l = l l a 0 l l and,hence a m = a Q . Now, a = c J d m 3 A a , = a^dm^ A. . . A a ^ d l ^ A a = a . Ther e f o r e , Tn o m-1 o o o -a = a^ d f f l ' ' A . . . A a £ d l ^ A a which i s i n G ( a Q ) , i . e . , C = G(a ) = G(M(C)) , as r e q u i r e d . Q.E.D. The element M(C) completely determines the c l o s e d group C . I f a € P(k) i s such that a = M(C) f o r some c l o s e d subgroup \ C of P(k) , then we say th a t a i s ge n e r a t i v e ( f o r the group C) , and we denote C by [a] . 1 . 8 P r o p o s i t i o n : a e P(k) i s g e n e r a t i v e i f and o n l y i f M(G ( a ) ) = a . Proof: Suppose a i s g e n e r a t i v e j then a = M(C) , f o r some c l o s e d subgroup C . Then, M(G ( a ) ) =M(G(M(C))) = M(C) = a . 1 4 . (By P r o p o s i t i o n 1.7-) Then, we have set a n a t u r a l 1 - 1 correspondence between the c l o s e d subgroups and the g e n e r a t i v e elements of P(k) . Consequently we have a n a t u r a l 1 - 1 correspondence between the g e n e r a t i v e elements of P(k) and the type-k. t r a n s l a t i o n - i n v a r i a n t groups. L e t a be g e n e r a t i v e i n P(k) ; then we denote by the t - i group c o r r e s p o n d i n g to the c l o s e d subgroup C = [a] under the r e p r e s e n t a t i o n map 3 , i . e . , H = H a i f 3(H) = C = [a] . L e t a be g e n e r a t i v e i n P(k) . Then, we d e f i n e the permutation a as f o l l o w s : f> l+k l+2k) i f a = 0 a = \ (a^ a^+k)...(a t a^+k) i f a = {a^,...,a^} L e t H = H t then we have the f o l l o w i n g : 1 . 9 P r o p o s i t i o n : H i s completely determined by a . In f a c t i f T e H , then T = a 1 ...a n J f o r some i n t e g e r s i ^ , . . . , I n where n >_ 1 . Proof: L e t H' be the set of a l l the a . . .a - ± r-- ' s ; then H' i s e a s i l y seen to be a t r a n s l a t i o n - i n v a r i a n t group. L e t 3 be the r e p r e s e n t a t i o n map, and c o n s i d e r #(H') = { 3 ( a [ i l ] . . . a C i n 3 ) } = { 3(a )A-• • A 3 ( a [ i n ] ) } = {( 3(a )) A. . . A ( 3 ( a ) ) ^ ± n h . But, by d e f i n i t i o n of a , :3(C T) = a . 15. Therefore, 3(H') = {a [ il?A. . : A c J 1 ^ } , hence 3(H') = [a] i . e . H' = H a Q.E.D. i L e t t i n g H = H a , for some generative element a , we c a l l the permutation a defined above, the generator of H j and we denote H by [a] . We extend t h i s d e f i n i t i o n to the t r i v i a l t - i group, by l e t t i n g a = 1 be i t s generator. In concluding t h i s section, we sh a l l give a simple i l l u s t r a t i o n of the above concepts. Let k = 3 , then i t w i l l be shown l a t e r that the generative elements of P(3) are: a± = {1} , a2 = {1,2} , = {1,2,3} and = 0 . Let = [CK ] , 1 <. i <. 4 , denote the corresponding type-3 t r a n s l a t i o n - i n v a r i a n t groups. Recall that a type -3 group H has Z1,Z2,Z-^ as the orbits f o r i t s action, where Z^ = {i, i+3, i+6j. . .} , 1 <. i <. 3 , and each r e H can be written as T i T 2 T 3 where acts on the integers i n Z^ . Now, = [CJ-J where the generator i s the transposition (l4). i s c l e a r l y the group 3^(3) • H 2 = [a 2J , where a 2 = (l4)(25) . Then H g consists of a l l permutations T = T]_T2T3 € S c o ^ ^ 3 f o r w n i c n either none or exactly two of the T i j T 2 ; , ' r 3 a r e o d d perm-utations . 16. = [a^] , where a ? = (14)(25)(36) . Then consists of a l l T = T2. T2 T3 e ^co(^) * ^ o r w n i - c n either none or a l l the T i > T 2 , T 3 a r e o d d permutations. = t a4^ » where = (1^7) . Then i s just A J 3 ) • CHAPTER 2 We have shown i n the l a s t chapter the e x i s t e n c e of a n a t u r a l 1-1 correspondence between the type-k t r a n s -l a t i o n - i n v a r i a n t groups and the c l o s e d subgroups ( e q u i v a l e n t l y , the g e n e r a t i v e elements) of P(k) , f o r each k >_ 1 . We s h a l l now examine the c l o s e d subgroups l a t t i c e o f P(k) i n g r e a t e r d e t a i l . 1. P r i n c i p a l and Minimal Subgroups. L e t k >_ 1 be f i x e d . L e t C be a c l o s e d subgroup of P(k) ; then C = [a] where a i s g e n e r a t i v e and any 3 e C has a.repres-r • i r * i e n t a t i o n (not unique) as a 1 A...Aa n J . Such a r e p r e s -e n t a t i o n f o r P i s s a i d to be of l e n g t h n . By an even (resp. odd) r e p r e s e n t a t i o n f o r (3 we mean a r e p r e s e n t a t i o n of even (resp. odd) l e n g t h . Given C , l e t C' = {(3 A / p e C} . Then, the c l o s e d subgroup C' i s c a l l e d the p r i n c i p a l subgroup o f C . 2.10 P r o p o s i t i o n : L e t C' be the p r i n c i p a l subgroup of a c l o s e d C . Then C' = [F3 e C / p has an even r e p r e s e n t a t i o n } . 18. Proof: L e t y e C' . Then Y = 3 A B J where 3 e C . L e t a ^ l ^ A. . . A a ^ n ^ be any r e p r e s e n t a t i o n of 3 , then Y = 3 A p C l ] = ( a [ i l ] A . . . A a [ i n ^ ) A ( a [ i l ] A . . . A a [ i n ^ ) ^ - = a''11'' A. . . A c ^ 1 ^ A a ^ l + 1 ^ A. . . Aa^n 4" 1^ which i s a r e p r e s e n t a t i o n of even l e n g t h 2n . Conversly, l e t a ^ l - ' A. . . A a ^ n ^ be an even r e p r e s -e n t a t i o n f o r y £ C . Now, i f Y ^ 0 , we can assume the i n t e g e r s i ^ , . . . , i to be d i s t i n c t , f o r we can e l i m i n a t e any p a i r of equal i n t e g e r s without changing the p a r i t y of the r e p r e s e n t a t i o n . Assume i ^ < . . . < i n , and d e f i n e 3 by 3 = ( a [ i l W i l + 1 ] A . . . A ^ A ( a ^ l n - 1 - ' A a ^ l n - 1 + i b ^ A . . . AcJ 1 1 1"" 1^) . Then, 3 e C , and c l e a r l y , 3 A 3^"^ = Y > i . e . , Y e C , hence C' = {3 e C / 3 has an even r e p r e s e n t a t i o n } . Q.E.D. Le t C Q be any c l o s e d subgroup and i t s p r i n c i p a l subgroup. Define, i n d u c t i v e l y , C j _ + ] _ t o be the p r i n c i p a l subgroup of • C\ , i >_ ° • Then we get a descend-i n g c h a i n of c l o s e d subgroups C Q > C-^  > . . . > C n > . . . . Hence, e v e n t u a l l y C n = C n f o r every n >_. n , f o r some n Q >_ o . o 2.11 D e f i n i t i o n : L e t C be c l o s e d and C' be i t s p r i n c i p a l subgroup. Then, C i s s a i d t o be minimal i f C == C' . Suppose C = [a] i s minimal} then a i s s a i d to be a minimal (generative) element. We note that C = 0 i s 19. minimal and that a c l o s e d C i s non-minimal i f f C' < C . Hence, s t a r t i n g w i t h a c l o s e d C we have a s t r i c t l y descending c h a i n 0 o > >...> C n where i £ the p r i n c i p a l subgroup of C i , o <_ i < n and C n i s m i n i m a l . i Before p r o c e e d i n g f u r t h e r w i t h the c h a r a c t e r i z a t i o n of non-minimal groups, we need the f o l l o w i n g r e s u l t s : 2.12 Eemma: {l , 2,...,k} i s the unique g e n e r a t i v e element a i n P(k) such t h a t ||a|| , k ... Proof: The group C = [0,{1,2,...,k}} i s c l e a r l y c l o s e d , and C = [ [1 ,2 ,...,k}] , i . e . , {l , 2,...,k} i s g e n e r a t i v e . Suppose t h a t ||a|| = k f o r some g e n e r a t i v e a i n P(k) , and r e c a l l t h a t a i s then the unique element of C f o r which ||a|| = k . Consider a A cJ"*"-' , an element of [a] . Then, k = ||a||;-< ||a A a-1-J,| < k+1 , i . e . , ||a A a-1- || i s e i t h e r k or k+1 . Now, i f j[ot A oJ 1-'|| = k , then '< a A oJ"^ = a , hence a = 0 which i s i m p o s s i b l e , as [|a|| = k . Then ||a A aJ 1"'|| musl: be" k+1 , i.e.', a A a'-1-' = 0 and as a ^ 0 , i t f o l l o w s that a = {l , 2,...,k} . Q.E.D. 2.13 Remark: In view of the above, we make the f o l l o w i n g remark: 20. I f a jL 0 i s g e n e r a t i v e i n P(k) , then ||a A a^-|| = ||a|f + 1 . '•• For, i f ||a|| < k , then i t i s always t r u e t h a t |fot A a'-1-'!! = ||a|| + 1 . Now.if | | a l l = k > t n e n a = {l,2,...,k} , hence a A a'-1-' = 0 and | | 0 | | by d e f i n i t i o n i s k+1 , i . e . , ||a A = | | a l l + 1 f o r a 1 1 g e n e r a t i v e a £ 0 ,.' 2.1k P r o p o s i t i o n : L e t C = [a] be a n o n - t r i v i a l , c l o s e d subgroup of P(k) . Then, the f o l l o w i n g are e q u i v a l e n t : (a) C i s non-minimal (b) C has a c l o s e d subgroup of index 2 . (c) a A a'-1-' i s g e n e r a t i v e i n P(k) . (d) Any two r e p r e s e n t a t i o n s o f 3 e C have the same p a r i t y . Proof: (aJ -Kb): L e t C' be the p r i n c i p a l subgroup of C = [a] . Then C' < C , hence, a | C' . But 3 e C i s i n the co s e t c' A a i f and o n l y i f 3 has an odd r e p r e s e n t -a t i o n i n C (see P r o p o s i t i o n 2 . 1 0 ) , i . e . , the c o s e t s C' and C' A a exhaust C , hence [C : C'] = 2 . (b)-*(c): L e t c' be a c l o s e d subgroup of index 2 i n C = [a] . Then a | C' and a A a ^ i s e i t h e r i n the c o s e t C' or C' A a . But, i f a A a'-1-' € C' A a , then a A a ^ = 3 A a , 3 e C' , so a'-1-' = 3 . Then, a a n d consequently a , are i n C' , which i s 21. i m p o s s i b l e ; i . e . , a A a--"- e C 7 . We s h o w t h a t C' I s , i n f a c t , [ ( a A a ' - 1 ^ ) ] . L e t 3 e C 7 . T h e n , a s 3 =}= a i t f o l l o w s t h a t | | a | | < | |3 | | . N o w , b y R e m a r k 2.13, w e h a v e | | a A a-1- || = | | a | | + 1 £ f o r e v e r y 3 e C ' , I . e . , a A c J 1 - ' i s a g e n e r a t i v e e l e m e n t f o r c' . ( c ) - * ( a ) : S u p p o s e a a n d a A oJ1-' a r e g e n e r a t i v e e l e m e n t s . T h e n , a s a ^ 0 , s o [ ( a A o J 1 - ' ) ] < [ a ] L e t C ' b e t h e p r i n c i p a l s u b g r o u p o f C = [ a ] . T h e n , y e C' =4 y = 3 4 , f o r s o m e 3 e C . S u p p o s e 3 = a 1 A . . . A c r l n J ; t?ien ( a f t e r r e a r r a n g e -m e n t o f t h e t e r m s i n t h e p r o d u c t ) 3 A 3^"^ = ( a A a [ l ] ) C i l ] A . . . A ( a A a C l ] ) [ i r J } 1 < e . 3 y = 3 A 3 [ l ] e [ ( a A oJ 1 - ')] , h e n c e , C ' < C a n d C i s n o n - m i n i m a l . ( a ) - » ( d ) : L e t C = [ a ] a n d C ' b e i t s p r i n c i p a l s u b -g r o u p . N o w , i f C i s m i n i m a l , i . e . , C 7 = C , t h e n a ( w h i c h i s a n o d d r e p r e s e n t a t i o n f o r a i n c') h a s a n e v e n r e p r e s e n t a t i o n a s w e l l ( b y 2.10). . C o n v e r s l y , s u p p o s e t h a t s o m e 3 € C h a s t h e r e p r e s e n t -a t i o n s c J x l ^ A . . . A a J 1 ^ a n d c J A : . . . A c J J n J w h e r e n i s e v e n , m i s o d d . T h e n , ( a ^ l ^ A . . . A c J i r i - ' ) A ( a C j ' l ] A . . . A [ j m ] ) = 0 , i . e . , a [ i l ] = a [ i 2 ^ . . . . . . A a ' - i n - ' A a ^ ' l - ' A . . . A a J ^ w h i c h i s a n e v e n r e p r e s e n t -a t i o n ( o f l e n g t h n+m -1 ) f o r a-1!- . T h e n , a s a = ( a ^ 1 l ^ ) ' • k ~ 1 l ^ a h a s a n e v e n r e p r e s e n t a t i o n , a s w e l l . T h e n , b y 2.10, a e C ' , h e n c e C ' = C a n d C i s m i n i m a l . Q.E.D. 22. Remarks: 1. I t f o l l o w s t h a t i f C = [a] I s non-minimal and C' i s the p r i n c i p a l subgroup of C , then C' i s , i n f a c t , [ ( a A a [ l ] ) ] . 2. L e t C = [a] . Then by 2.14(d), to show that C i s minimal, i t s u f f i c e s to f i n d an even r e p r e s e n t a t i o n f o r a i n C , or, e q u i v a l e n t l y , an odd r e p r e s e n t a t i o n f o r 0 I n C . 3. I f C i s non-minimal, then we can d e f i n e f o r 3 e C , I " fl If & has an even r e p r e s e n t a t i o n i n C . H((3) =\ -1 i f 3 has an odd r e p r e s e n t a t i o n i n C . Then, 2.14(d)'guarantees t h a t 3 i s w e l l - d e f i n e d . C l e a r l y , such ' p a r i t y ' f u n c t i o n can not be d e f i n e d f o r C minimal . 2. P r i n c i p a l Chains. We have shown i n the p r e v i o u s s e c t i o n that s t a r t i n g w i t h a c l o s e d subgroups C q , we can b u i l d a s t r i c t l y descend-i n g c h a i n of c l o s e d subgroups C Q > C^ >...> C n , where i s the p r i n c i p a l subgroup of C^ , o _< i < n , and C n i s minimal. I t i s n a t u r a l to ask whether a g i v e n c h a i n may be extended, i . e . , does there e x i s t a c l o s e d subgroup C whose p r i n c i p a l subgroup i s C Q ? Suppose C Q = [ a Q ] j then by the remarks p r e c e d i n g 2 . l 4 , i t i s e q u i v a l e n t to a s k i n g whether th e r e e x i s t s a g e n e r a t i v e element a , such that a A = a ? 23. L e t a e P(k) and l e t (a| = number of elements i n [ 1 i f ;|ct| i s even the set a . L e t e(a) = J . Then, [ - 1 i f |a| i s odd the answer to the above q u e s t i o n may be found i n the f o l l o w i n g : 2 . 1 5 P r o p o s i t i o n : L e t a be g e n e r a t i v e i n P(k) ,where e(a) = 1 . Then, t h e r e e x i s t s a unique p i n P(k) such t h a t 1 e P and fi A = a . Furthermore, P i s gener-a t i v e . Proof: L e t a be g e n e r a t i v e i n P(k) , w i t h e(a) = 1 . Now, i f a = 0 , then l e t p = ( 1 , 2 ,...,k} . By 2 . 1 1 , fi i s g e n e r a t i v e and c l e a r l y p A = a = 0 . Hence, assume a / 0 . Then we can w r i t e a as [ 1 , a 2,a^,a^,. . . , a 2 m _ 1 , a 2 m } f o r some m _> 1 , and 1 < a 2 <•. .. < a 2 m " D e f i n e p as [ 1 , 2 , . . . , a 2 ~ l , a ^ , a ^ + l , . . . , a ^ - l , . . . , a 2 m - l + 1 * ' ' ' , a 2 m - 1 ^ ' The*1 1 € P > a n d c l e a r l y p A p'-1-' = a To show t h a t p i s g e n e r a t i v e , assume that f o r some Y € G(p) (the c l o s e d subgroup generated by p ) , | | Y | | < l lPlI • Now, as y e G(p) , then y = p A . . . Ap - X n -f o r some i - . , . . . , i , and y A y - 1 - = (p A ) A. . . A(p A p [ l ] ) [ i n = a [ i l ] A . . . A a [ i n ] . i > e > i y A Y C l ] e [a] . But, as ||Y | | < k , we have ||Y A Y ^ || = | | Y ( | + 1 1 IIP || < ||a|| , which i s i m p o s s i b l e as a i s g e n e r a t i v e . Hence, i f y e G(p) , then ||p|| £ | | Y | | , i . e . , p i s gener-a t i v e . 2\. F i n a l l y , to show P unique, assume that there e x i s t s y e P(k) , such that l e y and y A y'"1"' = a . Then, Y A y ^ = M P^ 1 " 1 so (3 A y ) = ( P A y)-1- . Hence, e i t h e r p A y = 0 or p A y = {l , 2,...,k} . But 1 e P , 1 e y 1 | p 4 y and, t h e r e f o r e , - p A y must be 0 , i . e . , p = y , hence uniqueness i s proven. Q.E.D. Le t C Q = [a ] be a c l o s e d subgroup, where e ( a Q ) = Then, by s u c c e s s i v e a p p l i c a t i o n of P r o p o s i t i o n 2.15 we get a s t r i c t l y ascending c h a i n of c l o s e d subgroups C Q C ^ <...< C where f o r every -m _< i < o , c - j _ + i i s t n e p r i n c i p a l subgroup of C\ . Now, i f the c h a i n terminates a t c m = t a ] > then e(a) = -1 f o r otherwise we c o u l d extend the c h a i n even f u r t h e r . 2.16 D e f i n i t i o n : The c h a i n C > > . . . > C n of c l o s e d subgroups of P(k) such that: ( i ) C Q = [ a Q ] and 6 ( a Q ) = -1 ( i l ) C j _ + ] _ i s the p r i n c i p a l subgroup of , o _< i < n ( i i i ) C n i s minimal i s c a l l e d a p r i n c i p a l c h a i n of l e n g t h (n+l) i n P(k) ,. Example: L e t k = 8 , and c o n s i d e r ag = 0 . Then ag i s a minimal g e n e r a t i v e element i n P ( 8 ) and e(ag) = l . Then, by s u c c e s s i v e a p p l i c a t i o n of 2 .16, we get the f o l l o w i n g 25 . g e n e r a t i v e elements of P(8.) : a 8 = 0 , ay = [1,2,...,8} , a 6 = { 1 , 3 , 5 , 7 } a 5 = {1,2,5,6} , a 4 = {1 ,5} , a_. = {1 ,2 ,3 ,4} , a 2 = {1,3} , a1 = {1 ,2} , a Q = {1} . Now, as e ( a Q ) = -1 , we have gone as f a r as p o s s i b l e , hence we have a p r i n c i p a l c h a i n of l e n g t h 9 i n P(8) : [c_ 0] > [a^] >...> [ctg] , where [ a.^  +_. ] i s the p r i n c i p a l sub-group of [a±] , o <_ i < 8 . L e t PL be the type - 8 t r a n s l a t i o n T i n v a r i a n t group cor r e s p o n d i n g to [a^] , o <_ i <_ 8 . Then, we have S J 8 ) = H o > %>•••> % = A J 8 ) > where each i s a (maximal) subgroup of index 2 i n H ± , o <_ i <_ 8 . F i n a l l y , to r e c a l l the connect i o n between and , c o n s i d e r , f o r example, a = = { 1 , 2 , 5 , 6 } . Then the generator of H = i s the permutation (1 9) (2 10) (5 13) (6 14) . 2.17 Remarks: As a consequence o f the d e f i n i t i o n o f p r i n c i p a l c hains o f P ( k ) , we make the f o l l o w i n g o b s e r v a t i o n s : (a) Every c l o s e d subgroup C i n P(k) l i e s on one and only one p r i n c i p a l c h a i n i n P(k) . (b) Given a minimal subgroup C = [a] , the p r i n c i p a l c h a i n ending i n C can be e n t i r e l y determined, u s i n g 2 . 1 5 , 26. i . e . , the minimal subgroups ( e q u i v a l e n t l y , minimal g e n e r a t i v e elements) determine a l l the c l o s e d sub-groups o f P(k) . Le t C > C , > . . . > C be a p r i n c i p a l c h a i n i n P(k) , o 1 n c r v ' ' •where CL = [a^] , o_< i _< n . Now, suppose Y e . Then Y = M f o r some (3• e C Q . Hence, e ( Y ) = e(P A p [ l ] ) = e(f3) • e ( ( 3 [ l ] ) = (e ( 3 ) ) = 1 . Furthermore, € ( a Q ) = - 1 * hence aQ 4 , i . e . , ^ C q , and the l e n g t h of a p r i n c i p a l c h a i n i s a t l e a s t 2. Now, the g e n e r a t i v e elements s a t i s f y H a i + ] J I = ||au || + 1 f o r every o < i < n (see 2 . 1 3 ) . T h e r e f o r e , the l e n g t h of the p r i n c i p a l c h a i n i s Han'H - ||cto|| + 1 . Now, as 1 < H a 0 H a n d l f a n ^ <_ k + 1 , we have that the l e n g t h of the p r i n c i p a l c h a i n i s <_ (k+l) - 1 + 1 = k+1 i . e . , The l e n g t h of a p r i n c i p a l c h a i n i n P(k) i s a t l e a s t 2 and at most k+1 , f o r every k >_ 1 . Suppose a i s g e n e r a t i v e i n ; P(k^) and i n P(k^) , k l ^ k 2 * Now, i f a 4= 0 > then the method of 2.15 i s independent of .k , i . e . , C Q > > . . . > C = [a] i s a p a r t of a p r i n c i p a l c h a i n i n both P(k^) and P ( k 2 ) . However, the p r i n c i p a l c h a i n i n P(k^) and P(k^) s t a r t i n g w i t h C Q > > . . . > C = [a] need not be the same. For example, a = {1,2,3,4} w i l l be shown to be g e n e r a t i v e i n P(4) and i n P(12) , and the p r i n c i p a l c h a i n c o n t a i n i n g a i s : i n P (4) : [1] > [{1,2}] > [{1,3}] > [ {1 ,2 ,3 ,4} ] > 0 i n P (12 ) : [1] > [{1,2}] > [{1,3}] > [ {1 ,2 ,3 ,4} ] > [ {1 ,5} ] . 27 . 3. On c l a s s e s of g e n e r a t i v e elements. Before p r o c e e d i n g w i t h our main r e s u l t r e g a r d i n g the p r i n c i p a l c hains i n P(k) , we devote a s e c t i o n to methods of l o c a t i n g g e n e r a t i v e elements. We f i r s t show that the g e n e r a t i v e elements of P(k - J determine a c l a s s o f g e n e r a t i v e elements i n P ( k 2 ) i f d i v i d e s k 2 (k-^/k^). Le t k^ >_ 1 and k2=dk-^ f o r some d >_ 1 . Consider the map P(k- L) - P(kg) sending a = {a^,...,^} e P ( k 1 ) i n t o a* = [a*,...,a*} e P ( k g ) , where ' i * = 1 + ( i - l ) d f o r every 1 <_ i k^. Set 0 * = 0 . Then the f o l l o w i n g p r o p e r t i e s a re immediate f o r every a,p e PCk-J : * 1 . ||a|| < ||p|| i f and o n l y i f ||a*|| < ||p*|| . ' * 2 . ( a 1 - 1 ^ ) * = ( a * ) ^ 1 ^ , f o r every i n t e g e r i ... Then, the f o l l o w i n g i s t r u e : 2 .18 P r o p o s i t i o n : a i s g e n e r a t i v e i n P ( k 1 ) i f and o n l y i f a* i s g e n e r a t i v e i n P ( k 2 ) . Proof: Suppose t h a t a i s g e n e r a t i v e i n p ( k ^ ) , and c o n s i d e r the c l o s e d subgroup generated by a*, G(a*) i n P ( k 2 ) . L e t G(a*) = [p] , where p e P ( k 2 ) . We show that p = a* . Now, i f p = 0 , then G(a*) = [p] = 0 , hence a* = 0 = P , and we are f i n i s h e d . 28 . Assume 3 ^ 0 . Then as 3 6 G(a*) , we may w r i t e 3 as (a*) ^"""lA. . . A (a*) l n w i t h i - . 5 i 2 , . . . , i n d i s t i n c t . L e t S = {i . / i . = 1 mod (d) , 1 <_ j <_ n} . I f S = 0 , and as l e a * , i t f o l l o w s that 1 | (a*)'- iJ-' f o r every 1 < j <_ n , hence 1 «• p which c o n t r a d i c t s that 3 i s ge n e r a t i v e . T h e r e f o r e , S ^ 0 , and suppose that S = {JV * ' •'V • L e t y = ( a * ) [ j ' l ] A . . . A(a * ) [ j ' m 3 _ T h e n Y ^ 0 , and c l e a r l y ||Y|| <. | |3 | | <_ Ho1*|| • Furthermore, Y = (a-j_)* , f o r some e P ^ ) . But then, | | ( a 1 ) * | | _< | |o*|| , hence by , Ha-jJI <. ||a|| . As a i s g e n e r a t i v e , we must have = a , and t h e r e f o r e , 3 = a*, as r e q u i r e d . Conversly., suppose t h a t a* i s g e n e r a t i v e i n P ( k p ) . L e t B = {(a*)^l^A.'... A ( a * ) ^ n 3 / ± 0 m o d ( d) s j = 1 , 2 , . . . ,n , n >_ 1} . Then, B i s a subgroup (not n e c e s s a r i l y c l o s e d ) of [a*] , and i f Y e B , then f o r every n ^ y ^ d ] € -g a n c j f u r t h e r m o r e , Y = 3 * f o r some 3 e P(k ]_) . L e t C = (3 e P ^ ) / 3 * e B} . Now, i f 3 e C , then by * 2 . , O ^ ) * = ( 0 * ) ^ e B , hence £ ^ e C . The r e f o r e C i s c l o s e d i n P(k^) , and by * 1 . , we c l e a r l y must have C = [a] . But then a i s g e n e r a t i v e i n P(k-j_) • Q.E.D. Le t r > 1 and d >_ 1 be i n t e g e r s and l e t a ( r , d )  be the set [1 , 1+d,...,!+(r -1)d] . 29. 2 .19 P r o p o s i t i o n : a (2,d) i s g e n e r a t i v e i n P(k) i f and on l y i f d|k and 2d <_ k . Proof: Suppose that d|k and 2d £ k . Then k = 6k^ and k-j_ >_ 2 . Hence {1,2} i s g e n e r a t i v e i n P ( k 1 ) , and by 2 . 1 8 , {1,2}* = {1, 1+d} i s ge n e r a t i v e i n P ( k 1 d ) , i . e . , a (2,d) i s g e n e r a t i v e i n P(k) . To show n e c e s s i t y , suppose that a = a ( 2,d) = {1,1+d} i s ge n e r a t i v e i n P(k) . -Now, i f 2d > k , then a A o J ^ = {1, (l+2d) mod k}. But, as (l+2d) mod k '< 1+d , then J| ct A c t ^ || < ||a|,| which i s a c o n t r a d i c t i o n , i . e . , 2d must be <_ k . Assume that d | k . Then, k = d+d 1 , where dj £ md f o r any m >_ o . Now, c J d l - ' = {(1+djO mod k, (l+d+d^mod k} = {1, l + d 1 ) , and as (|a'-d l-'|| > ||a|| , we must have d 1 > d . L e t d-j_ = md + r where m >_ 1 , and o < r < d . L e t p = a A c J d W 2 d ] A. . . Aa C = {1, l+d}A{l+d, l+2d}A...A{l+(m-l)d, 1+md} = {1, 1+md} . Then, Y = a ^ 1 ^ p = {1 + md, l+c^} = {1+md, 1+md+r} , and t h e r e f o r e , Y''~ m d'' = {1, 1+r} e [a] . i . e . , ||Y [" m d ] || = r+1 < d < Ifcx11 , which i s im p o s s i b l e . Therefore d must d i v i d e k . Q. E. D. 30. 2.20 C o r o l l a r y : a ( r , d ) i s g e n e r a t i v e i n P(k) i f rd|k . Proof: Suppose r|k and assume d = 1 , Now, i f r = k . then a ( r , l ) = {l , 2,...,k} which we have shown to be gener-a t i v e i n P(k) . Assume, then r < k . Then 2 r <_ k , and by 2 .19 (where r r e p l a c e s d ) , {1, 1+r} i s gener-a t i v e i n P(k) . But then, by the method of 2 . 1 5 , we know tha t { l , 2 , . . . , r } i s g e n e r a t i v e i n P(k) . i . e . a ( r , l ) i s g e n e r a t i v e i n P(k) , whenever r|k . Assume, then, rd|k . i . e . , k = dk^. and r|k^ . But by the above a ( r , l ) = { l , 2 , . . . , r } i s g e n e r a t i v e i n P(k- L) , and then, by 2 . 1 8 , { l , 2 , . . . , r } * i s g e n e r a t i v e i n P ( k 1 d ) . But { 1 , 2 , ...,r}* = { 1 , 1+d,,..,l+(r-l)d} = a ( r , d ) . Q. E.D. Remark: Suppose C = [a] i s a minimal subgroup. Then, a has an even r e p r e s e n t a t i o n i n C . In f a c t , a may be w r i t t e n as (*) a = (a A a'-1-') A. . . A(a A a ' - 1 ^ ) ' - 1 ^ } f o r some ^ I ' - ' - ^ n ' Now, c e r t a i n c l a s s e s of g e n e r a t i v e elements can be o b t a i n e d by ' s o l v i n g ' r e l a t i o n s i n a such a s r ( * ) , i . e . , i n the s i m p l e s t p o s s i b l e case of (*) a = a A , hence a = 0 which i s always a (minimal) g e n e r a t i v e element i n P(k) . The next s i m p l e s t case of (*) Is a = (a A a J 1 ' ' ) . The c l a s s of g e n e r a t i v e elements s a t i s - , f y i n g t h i s r e l a t i o n can be shown to be g i v e n completely by a = {1 ,2 ,4 ,5 ,...,l+3m, 2+3m} i n P(k> , where k = 3(l+m) 31. and m = 0 , 1 , 2 , . . . . To i l l u s t r a t e , l e t k = 12; i . e . , k = 3(l+m) where m = 3 . Then, a = { 1 , 2 , 4 , 5 , 7 , 8 , 1 0 , 1 1 } i s a minimal g e n e r a t i v e element i n P ( l 2 ) . The e n t i r e p r i n c i p a l c h a i n i n P ( l 2 ) can now be computed, i . e . , i t i s [ { 1 , 2 ,4 ,6 ,7 } ] > [ { 1 , 3 , 4 , 5 , 6 , 8 } ] > [ { 1 , 2 , 3 . 7 , 8 , 9 } ] > [ {1 ,4 ,7 ,10} ] > [ { 1 , 2 , 4 , 5 , 7 , 8 , 1 0 , 1 1 } ] . T h i s technique gets r a t h e r i n v o l v e d when a s a t i s f i e s r e l a t i o n s of 'higher order'. We mention, i n pass-i n g , t h a t the f i r s t s o l u t i o n . t o the r e l a t i o n a = (a A oJ'^)^ i s g i v e n by a = {1 ,3,4,5} i n P (7) . 4. The l e n g t h of a p r i n c i p a l c h a i n i n P ( k ) . The methods f o r l o c a t i n g g e n e r a t i v e elements d e s c r i b e d i n 3- do not g i v e a l l the g e n e r a t i v e s of P ( k ) . However, f o r k s m a l l , the ge n e r a t i v e elements not found by the u s u a l methods, may be e a s i l y computed. We s h a l l i l l u s t r a t e the complete l a t t i c e o f the c l o s e d subgroups o f P(k) , f o r 1£ k <_ 9 • Looking a t the l a t t i c e f o r these f i r s t few va l u e s o f k w i l l s u f f i c e i n g i v i n g us a p i c t u r e o f the l a t t i c e f o r any P(k) . Let a,p be ge n e r a t i v e s i n P(k ) . Then by [a]- -[(-] we s h a l l mean that [a] and J}3] are s u c c e s s i v e terms o f a p r i n c i p a l c h a i n , i . e . , [fJ] i s the p r i n c i p a l sub-group o f [a] . By [a] we s h a l l mean [a] > [p] . [p] 32 . L e t 1 be the g e n e r a t i v e element a = {1} , i.e., [ l ] = P(k) . Then we have the f o l l o w i n g : k = 1 [1] [0] k = - _ 2 . : [1]. [ { 1 , 2 } ] [0] [13 - [ { 1 , 2 } ] k = 3 : [ { 1 , 2 , 3 } ] [0] k = 4 : [1] [ { 1 , 2 } ] [ { 1 , 3 } ] [ { 1 , 2 , 3 , 4 } ] [0] k = 5 [1]- • [ { 1 , 2 } ] [ { 1 , 2 , ...,5)]' [0] [1]- [ { 1 , 2 } ] -k = 6 : [ { 1 , 2 , 3 } ] - [ { 1 , 4 } ] -[ { 1 , 3 , 5 } ] [ { 1 , 2 , . : . , 6 } ] -- [ { 1 , 3 } ] [ { 1 , 2 , 4 , 5 } ] [0] [1] [ { 1 , 2 } ] k = 7 : [ { l , 3 ' , 4 } ] - _ [{1,2,5,5)] i U , 2 , 4 } ] . [ {i", 3 , 4 , 5 } 3 [ { 1 , 2 , : . : , 7 } ] - [01 33-k = 8 : [1] [ { 1 , 2 } ] [ { 1 , 3 } ] — [ { 1 , 2 , 3 , 4 } ] — [ { 1 , 5 } ] -— [ { 1 , 2 , 5 , 6 } ] [ { 1 , 3 , 5 , 7 } ] [{1 ,2 , . . . ,8}}— [0] . [1]_ [ { 1 , 2 } ] k = 9 : [ { 1 , 2 , 3 } ] [{1,4}'] [ {1 ,4 ,7} ] [ { 1 , 2 , 4 ^ , 7 , 8 } ] [ { 1 , 2 , 9} ] -10] The s t r u c t u r e o f the c l o s e d subgroups of P ( k ) , 1 <_ k <_ 9 , i s very s u g g e s t i v e . We may c o n j e c t u r e that a l l the p r i n c i p a l chains i n P(k) have the same l e n g t h . F u r t h e r -2 3 more, l o o k i n g at the cases k = 2 , 2 , 2 , we may c o n j e c t u r e t h a t there i s o n l y one p r i n c i p a l c h a i n i n P ( 2 n ) , f o r every n >_ 1 , whose l e n g t h i s 2 n + 1 . We w i l l prove, i n f a c t , t h a t every p r i n c i p a l c h a i n i n P(k) has l e n g t h 2n^^ + 1 , where n(k) i s the l a r g e s t i n t e g e r n >_ o f o r which 2 / k . L e t k >_ 1 be f i x e d , and l e t a be g e n e r a t i v e element of P(k) . By chn(a) we w i l l mean the p r i n c i p a l c h a i n i n P(k) c o n t a i n i n g the c l o s e d group [a] . By p e chn(a) , p a g e n e r a t i v e element of P(k) , we mean t h a t [p] l i e s i n chn(a) , i . e . , P e chn(a) i f and on l y i f chn(p) = chn(a) . Chn(l) , the p r i n c i p a l c h a i n i n P(k) whose f i r s t group i s [1] (=P(k)) w i l l be c a l l e d the upper c h a i n of P(k) . 34. Chn(0) , the p r i n c i p a l c h a i n i n P(k) whose l a s t group i s 0 w i l l be c a l l e d the lower c h a i n of P ( k ) . We f i r s t prove the f o l l o w i n g r e s u l t s : 2.21 P r o p o s i t i o n : A c l o s e d subgroup of a minimal group i s minimal. Proof: L e t [aJ be a minimal subgroup of P(k) , and suppose t h a t [0] < [a] f o r some g e n e r a t i v e element 0 . Now, as a i s minimal, then by 2.14(d) , a has an even r . -I r" 1 r e p r e s e n t a t i o n o r ^ l - 1 A. . . A o r J m J , m even, i n [a] . Now as p c [ a ] , 0 has r e p r e s e n t a t i o n as a^lh. . . 4a 1 1*' i n [a] . But then, 0 = ( a ^ l ^ A. . . AcJ ^  ) ^ A . . . A (a'" A. . . A a ^ m ^ ) . Rearranging the terms i n the product we have 0 = ( a ^ l ^ A . . . A a ^ r J ) .. A ( a [ i l ^ A. . . Aa 1- 1™- 1) r . -I r . -I = 0 "-"l A. . . A0 ^ m , i . e . , 0 has an even r e p r e s e n t a t i o n • i n [0] , and hence, [0] i s minimal. Q.E.D. 2.22 P r o p o s i t i o n : Every p r i n c i p a l c h a i n i n P(k) has the same l e n g t h . Proof: L e t c h n ( l ) = [a ] > [a±] >...>[an] ( i . e . , aQ = { l } ) , and l e t chn(0) = [0O] > [0-jJ >...> [0 J be an a r b i t r a r y p r i n c i p a l c h a i n i n P(k) . Assume f i r s t t h a t m > n . Now, as '[a ] = P(k) , we have [0Q] < [ a Q ] . But then, [01] = {0 A 0^^ / 0 e [0Q]} < {a A a [ 1 ] / a e [aQ]} , i . e . [0-.] < [a±] . 35. Proceeding i n d u c t i v e l y , we can show that [p j < [a ]^ f o r every o <_ i <_ n . But as [ a n ] i s minimal and as [ P n J < [ a n ] , i t f o l l o w s by 2.21 , t h a t [p ] i s minimal, c o n t r a d i c t i n g that m > n . Assume, then, t h a t m < n . Now, as [PQ] i s the f i r s t group o f the p r i n c i p a l c h a i n , then e(P D) = -1 • Hence, as aQ = {1} , PQ has a r e p r e s e n t a t i o n i n [ a Q ] as a j ^ l ^ A . . . A c J 1 ^ where t i s odd . Now, P x = pQ A p^ 1 3 (by 2.14(c)), hence ^ = ( a ^ l ] A . . . A a ^ t ] ) A ( a ^ 1 l ] A . . . A a j ^ t 3) ^ = (a. A c J l ] ) [ i l ] A . . . A ( a 4 a [ l ] ) [ l t ] = a j 1 ! 3 A . . . A a . . [ i t 3 , v o 1 ' v o o ' 1 1 . i . e . , Pj has an odd r e p r e s e n t a t i o n i n [a^] . Proceeding i n d u c t i v e l y , we have Q = a ] - 1 ! 3 A . . . A a [ l 1 : 3 . Now, as [ p i m m m ' m i s minimal, 0 has an odd r e p r e s e n t a t i o n i n [p ] (see 2.14(d)) r • I r • "1 S u b s t i t u t i n g a ^ x l A . . . A o r 1 f c J f o r Pm i n such an odd r e p r e s e n t a t i o n f o r 0 , we get an odd r e p r e s e n t a t i o n o f 0 i n [ a m ] > i . e . , [ a m J i s minimal, c o n t r a d i c t i n g m < n . Ther e f o r e m = n ./ Q.E.D. Le t chn(a) = [ a ] > [a^] > . . . > [ a n ] and chn(p) = [pQ] > [p-J >...>.-[g^] be any two chains i n P(k.) . Then we ca n - d e f i n e a p a r t i a l o r d e r i n g on the chains o f P(k) , by l e t t i n g chn(a) < chn(p) , i f and on l y i f , [a j < [P^l f o r every o <_ I <_ n . In the p r o o f of 2.22, we have shown that f o r any g e n e r a t i v e a , chn ( i a ) < c h n ( l ) . In f a c t , i t i s e a s i l y shown that chn(0) < chn(a) < c h n ( l ) . 36. We make now the f o l l o w i n g o b s e r v a t i o n : l e t a e P(k) , and d e f i n e , i n d u c t i v e l y , a. , = a. A aj- 1-' f o r o v 1+1 1 i i >_ o . Then, by s t r a i g h t - f o r w a r d i n d u c t i o n argument i t f o l l o w s t h a t a ~ = a A af-2*1-- f o r every n > o . Before gxx O O — p r o v i n g the theorem on the l e n g t h of a p r i n c i p a l c h a i n , we -need the f o l l o w i n g : 2.23 Lemma: L e t aQ e P(k) be such that 1 e a Q and ||a0H < k , and l e t p be g e n e r a t i v e i n P(k) . Suppose that 6 = a A a'-211-' where n > o i s such that 1 < 2 n < o o — — — - ||a H ). Then, a Q i s g e n e r a t i v e and a Q € chn(p) . Proof: D e f i n e i n d u c t i v e l y , a. _ = a. A a?1-- , o < i < 2 n . 1+JL 1 1 — Then, as remarked above, a ^ n = a Q A a ^ 2 ^ , hence a ^ n = (3 , which we are giv e n to be g e n e r a t i v e . Furthermore, l e a , and as 1 <_ 2 n < (k - ||aj|) , i t f o l l o w s that l e a , x f o r every o < I < 2 o Hence, we have the f o l l o w i n g s i t u a t i o n : a ^ n i s g e n e r a t i v e , and a ^ n = a^n-l A a ^ l i > where 1 e a 2 n _-j_ . Comparing w i t h 2.15, t h i s i m p l i e s that ^^x--\_ i - s g e n e r a t i v e and c l e a r l y , a ^ n - ± e c h n ( a 2 n ) . We proceed i n d u c t i v e l y . Given that a^+j_ i-s g e n e r a t i v e f o r some o < i < 2 n , then, as a. A a| 1-' = a. — i i 1+1 and 1 e a^ , i t f o l l o w s t h a t a^ i s g e n e r a t i v e , and e c h n ( a ^ + ^ ) = c h n ( a ^ n ) . In p a r t i c u l a r , a Q i s g e n e r a t i v e and a Q e c h n ( a ^ n ) = c h n ( P ) . Q.E.D. 37. L e t k >_ 1 be g i v e n and l e t n(k) denote the l a r g e s t i n t e g e r n > o , f o r which 2 n / k . 2.24 Theorem: Every p r i n c i p a l c h a i n i n R(k) has l e n g t h 2 n ( k ) + 1 m Proof: Suppose, f i r s t , t h a t k i s odd; i . e . , n(k) = o .. Then k = 2m+l , f o r some m > o . L e t C > C-. be> the — o 1 f i r s t two terms of a p r i n c i p a l c h a i n i n P(k) . We show that C-j i s minimal. L e t -C = [a ] and CL = [a, ] and 1 o o J 1 1 c o n s i d e r p = a'-1'3 A oJ2"' A. . . A cJ 2 m 3 e C . Now, as p O 0. O O i r has even r e p r e s e n t a t i o n i n C Q , then (2.10), p e C^ . Furthermore, p A p [ l ] = ( J l ] A. . . A a ' [ 2 m ] ) A ( a [ 2 ] A. . . A a [ 2 m + 1 ] ) v o o K o o = a ^ 1 3 A a ^ 2 m + l 3 but as 2m+l = k and a £ k 3 = aQ , we have P A p'-13 = a ^ 1 3 A a Q = ^ as [o^] i s the p r i n c i p a l sub-group of [aQ] . L e t p = a j ^ 1 " ' A... A a^~n^ be any r e p r e s e n t a t i o n ' f o r p^  i n [a^] . Then, s u b s t i t u t i n g t h i s e x p r e s s i o n f o r P i n cxj = P A p^x3 , we get an even r e p r e s e n t a t i o n f o r i n [a-jj , i . e . , = [a-J i s minimal (2.14(d)). Hence, I f k i s odd, every p r i n c i p a l c h a i n i n P(k) has l e n g t h 2 = 2 n ^ k ^ + 1 . Suppose then, t h a t k i s even, i . e . , n(k) > o . L e t r _> 1 be a f i x e d , but a r b i t r a r y odd number. Then, we want to show that every p r i n c i p a l c h a i n i n P(k) , where k = 2 n r , has l e n g t h 2 n + 1 , f o r every n > o . By 2.22, i t s u f f i c e s to l o o k at the upper c h a i n of P(k) . 38. L e t k = 2 n r f o r some n > o , and l e t a = [1,2,. . . ,2 m } f o r every o <_ m <_ n . Then, by 2 . 2 0 , am ^~ a ^ 2 m - ' 1 ^ i s g e n e r a t i v e i n P(k) f o r every o _< m _< n Claim: e c h n ( l ) f o r o <_ ra <_ n . I t s u f f i c e s to show t h a t a m and am+j_ lie:;on" the same c h a i n where o <_ m < n . As a G = {1} > i t w i l l f o l l o w t h a t they a l l l i e on the upper c h a i n . By the d e f i n i t i o n of a m ' s , i t i s immediate that Tom] a T = a A a , o < m < n . Furthermore, a .. i s m+1 m m 3 — 3 m+1 g e n e r a t i v e , 1 e affi (as ct m ^  0 i s g e n e r a t i v e ) , and 2 m ± (k - | | a m l l )= ( 2 n r - 2 m ) , i . e . , a m + 1 and a m s a t i s f y a l l the c o n d i t i o n s of 2 . 2 3 , hence am and a m + 1 l i e on the same (upper) c h a i n where o <_ m < n . In p a r t i c u l a r , a n = [ 1 , 2 , . . . , 2 n } e c h n ( l ) , hence (see. 2 . 1 3 ) , the l e n g t h o f the upper c h a i n i s a t l e a s t 2 n In P(k) , where k = 2 n r . Now, i f r = l , I . e . , k = 2 n , then a n A a£Hj=--0 which i s a minimal g e n e r a t i v e element. Hence, we have: Chn(l) = [1] > [{1,2}] > . . .> [ { 1 , 2 , . . . , 2 n } ] > 0 which i s of l e n g t h 2 n + 1 i n P( '2 n) , as r e q u i r e d . Now, i f r > 1 , l e t 0 = a n A a^ 1- 1 = {1, l+2 n } [ ] 'n Then, by 2 . 1 9 , 3 = a ( 2 , 2 n ) , w i t h 2 - 2 n < r< 2 n = k , hence, (3 i s g e n e r a t i v e i n P(k) and i s i n c h n ( l ) . Then, c h n ( l ) has a t l e a s t 2 n + l terms: [ l ] > [{1,2}] >...> [ { 1 , 2 , . . . , 2 n ] ] > [{l, l+ 2 n } ] , and the theorem w i l l be completed by showing 0 to be minimal. I 39. L e t p 1 = 3^2 n ] = ( ( 1 + 2 n ) m o d k, (1+2 -2 n ) mod k} = { l+2 n 5 1+2-2 n} P 2 = p ^ 2 ' 2 ^ = { ( i+2 -2 n ) mod k, ( l + 3 - 2 n ) mod k} = {1+2-2 n , 1+3-2 n} P ^ = pK*-1^ = { ( i + ( r - l ) 2 n ) m o d k ,(l+r•2 n ) m o d k } = { l + ( r - l ) 2 n , 1} s i n c e k = r - 2 r o n i C l e a r l y , p A (3^ A... A p ^ = 0 , i . e . , 0 = (3 A JA...A r ( r - l } 2 n l P v y J , which i s an odd r e p r e s e n t a t i o n f o r 0 i n [p] The r e f o r e , by 2.14(d), [p] i s minimal. Q. E.D. 2.25 C o r o l l a r y : There are p r e c i s e l y 2 +1 t r a n s l a t i o n -i n v a r i a n t groups o f t y p e - 2 n f o r every n >. o . Proof: As chn(0) = c h n ( l ) i n . P ( 2 n ) , there, i s o n l y one p r i n c i p a l c h a i n and hence, e x a c t l y 2 n + l c l o s e d subgroups i n P ( 2 n ) • By the 1-1 correspondence between the c l o s e d sub-groups o f P ( 2 n ) and the t y p e - 2 n t r a n s l a t i o n - i n v a r i a n t groups we get our r e s u l t . Q.E.D. 4 0 . We conclude t h i s chapter w i t h the f o l l o w i n g remark about the number of p r i n c i p a l chains i n P ( k ) , k _> 1 : Le t k >_ 1 be g i v e n , and l e t p ^ . ^ - p ] ^ b i the prime decomposition o f k , where k^ >_ 1 f o r every 1 <_ i <_ n . Le t d(k) denote the t o t a l number of d i v i s o r s of k , i . e . , d(k) = ( l + k j ) ( l + k 2 ) . . . ( l + k n ) . L e t n(k) denote as above the l a r g e s t i n t e g e r n such that 2 n / k . Consider any d i v i s o r d of k of the form 2 m.r , where r > 1 i s odd, and 1 <_ m <_ n(k) . Then, by 2.20, a(r,2 m) = {1, l+2 m,...,l+(r-l)2 m} i s g e n e r a t i v e i n P(k) , and as e(a(r,2 r n)) = -1 , the c l o s e d group [ a ( r , 2 m ) j i s the f i r s t term of chn(a(r,2 m)). Now, i f d j = 2 m l r ^ , r ^ > 1 , i s d i f f e r e n t from d , then, a ( r j , 2 m l ) i S a d i f f e r e n t g e n e r a t i v e element from a(r,2 m)' , hence c h n ( a ( r ^ , 2 m l ) ) i s d i f f e r e n t from chn(a(r,2 m)) . For the remaining d i v i s o r s of k , namely 2°,2x,...,2n the g e n e r a t i v e elements a ( l , 2 m ) e c h n ( l ) f o r every 0 £ m _< n(k) . Hence, there are at l e a s t ( d ( k ) - n ( k ) ) p r i n c i p a l c hains i n P(k) , f o r k _> 1 . CHAPTER 3 1. Application;;- A d m i s s i b l e sequences. We have mentioned at the b e g i n n i n g o f the p r e s e n t -a t i o n t hat the n o t i o n o f a t r a n s l a t i o n - i n v a r i a n t group was motivat e d by a c e r t a i n m u l t i l i n e a r - a l g e b r a problem. • We s h a l l i n t r o d u c e the problem now, and show how the t r a n s -l a t i o n - i n v a r i a n t groups a r i s e from i t . We fehall f i r s t r e c a l l some m u l t i l i n e a r - a l g e b r a concepts. Throughout t h i s chapter, l e t F be a f i x e d , but a r b i t r a r y f i e l d . Then, a l l v e c t o r spaces c o n s i d e r e d w i l l be over F . Le t G beua group; then by a c h a r a c t e r X of G , we mean a homomorphism X : G -* F* (where F* i s the m u l t i p l i c a t i v e group of the f i e l d F) . Given a v e c t o r space V , l e t V 0 1 be the c a r t e s i a n product o f m co p i e s o f V , f o r every m _> 1 . Le t S m denote the f u l l symmetric group on the l e t t e r s {1,2,...,m} . Le t G < , f o r some f i x e d m > 1 , and l e t X m ' — 3 be a c h a r a c t e r f o r G . L e t V and W be v e c t o r spaces . 42. 5.26 D e f i n i t i o n : A m u l t i l i n e a r map f : V 0 1 -• ¥ i s s a i d to be symmetric w i t h r e s p e c t to G and X i f f ( x x , v) c __ v a ( l ) 5 cr(m)< X ( a ) f ( X j , . . . 3 x ^ ) f o r every a € G , x ± e V . I f f i s symmetric w i t h r e s p e c t to G and X , we s h a l l say simply t h a t f i s a (G,X)-map. Le t f : V™ - W be a (G,X)-map; then: 3 .27 D e f i n i t i o n : We say t h a t W (together w i t h the map f ) i s a", symmetry c l a s s 1 of tensors over V a s s o c i a t e d w i t h  G and X , i f W has the f o l l o w i n g p r o p e r t y : F o r every (G,X)-map g of v"1 i n t o a v e c t o r space T , th e r e e x i s t s a unique l i n e a r map h : W -• T such that h-f = g j i . e . , such that the diagram f V"1 > W T commutes. I t has been shown (see [ 6 ] , [ 7 ] ) t h a t given the p a i r G and X , the symmetry c l a s s o f tensors over V e x i s t s and i s unique up t o isomorphism of v e c t o r spaces. Such a space i s u s u a l l y denoted (see [ 6 ] ) by V^(G) , and we s h a l l c a l l V^(G) simply the (G 3X)-space over V . 43. We note t h a t i f f : V™ - V^(G) i s the map as i n D e f i n i t i o n 3.27, then the v e c t o r s f ( x 1 , . . . , x m ) are denoted by xi*--'* xm > a n d they span V^(G) . The n o t i o n of a (G,X)-space over V g e n e r a l i z e s the well-known tensor, Grassmann and symmetric spaces, f o r an a p p r o p r i a t e c h o i c e of G and X , i . e . , 1. I f G = 1 and X = 1 , then a (G,X)-map Is simply a m u l t i l i n e a r map and V^(G) = <§V - the tensor -. space. In t h i s case the v e c t o r s •x~i*--'*xm a r e denoted by X j ® . . . ® ^ . 2. I f G = S m and X = e = " s i g n of the permut-a t i o n " c h a r a c t e r , then a (G,X)-map (where the f i e l d P i s such that charP ^ 2) i s a skew-symmetric map, and V^(G) = ^ (V) - the Grassmann space. The v e c t o r s x l * ' * ' * x m a r e u s u a H y denoted by X j A . . . A x m . 3. I f G = S m and X = 1 then a (G,X)-map i s a symmetric map and V^(G) = S m ( V ) - the symmetric space. . Now";, f o r ' each of the above ' c l a s s i c a l ' spaces we have a n a t u r a l c o n s t r u c t i o n of a r e g u l a r l y - g r a d e d a l g e b r a , namely, the t e n s o r , Grassmann and symmetric a l g e b r a . We" s h a l l i n d i c a t e the c o n s t r u c t i o n of the Grassmann a l g e b r a , the others b e i n g analogous: 44. For every m >_ 1 , l e t G m = S m and l e t X^ be t h e ' ' s i g n of t h e j p e r m u t a t i o n ' c h a r a c t e r . We assume t h a t c'harF t 2 . Then we have seen that A(V) - the Grassmann space, i s the ( G ^ X ^ ) - s p a c e over V , f o r every m _> 1 . S e t t i n g A ( V ) = F , we l e t A = (+) A ( V ) - the d i r e c t sum. m> o m, As the spaces A(V) i s unique up t o isomorphism f o r every m. . m >_ o , we may assume t h a t A = S A(V) - the i n t e r n a l m>o d i r e c t sum. q L e t B e ( J \ ( V ) x A ( V ) ) x P A Q ( V ) be the r e l a t i o n p > q g i v e n by B p , q = { ( ( X 1 A - - - A ^ p > y-j_A...Ay 2), x-A. . . AXpAy-^A. .-Ay q)} f o r every p > o , q > o Bp,o = ( ( ( x i A - - - A ^ : p , a) , a(x 1A...AX )} f o r every p > o , i . e . , Bp Q i s s c a l a r m u l t i p l i c a t i o n of the v e c t o r s of K(v) B Q i s d e f i n e d s i m i l a r l y f o r every q > o . F i n a l l y , B Q = [ ( ( a , p ) , ap)} , i . e . B Q i s j u s t the m u l t i p l i c a t i o n i n A ( V ) = F . Then i t i s well-known t h a t B i s a b i l i n e a r map V> q f o r every p >_ o , q >_ o . Hence, there e x i s t s a unique b i l i n e a r map B o AxA - A which extends a l l the B 's , i . e . , B / 5\(V) x 5.(V) = B f o r every p >_ o , q >_ o p, q 45. (see [5]) . D e f i n i n g m u l t i p l i c a t i o n i n A by x-y = B(x,y) where x,y € A , makes A a r e g u l a r l y - g r a d e d a l g e b r a over P . A i s , i n f a c t , the Grassmann a l g e b r a u s u a l l y denoted by A ( V ) . The c o n s t r u c t i o n of the t e n s o r a l g e b r a ®Y , and of the symmetric a l g e b r a S(V) i s analogous. Consider the f o l l o w i n g g e n e r a l s i t u a t i o n : Lefe V be a v e c t o r space over P . L e t G m < S m and l e t ^ be a c h a r a c t e r of G m f o r every m _> 1 . Then the space ) e x i s t s and i s m unique up to isomorphism. Set t o be a f i x e d , but a r b i t r a r y ( G m , X m ) -space over V f o r every m >_ 1 , where V^1-1 = V . Set v(°) = F , and d e f i n e A(V) = Q^V^ - the m>_o (m) d i r e c t sum. L e t i m : ' -» A(V) be the c a n o n i c a l i n j e c t -i o n s f o r every m >_ o . Since i m(V"^ m' 1) i s a l s o a f i x e d ( G m , X m ) - s p a c e over V f o r every m >_ 1 , and i Q(V^°^) = P , we assume t h a t A(V) = S y(m) _ ^he i n t e r n a l d i r e c t sum. m>o Then we c a l l the sequence of p a i r s ^ G r a' X m--m>l a d m i s s i b l e f o r V i f A(V) = S v ( m ) can be g i v e n a ( r e g u l a r l y - g r a d e d ) m>o' •. a l g e b r a s t r u c t u r e , where m u l t i p l i c a t i o n i s d e f i n e d by (x^*. . .*Xp)- (y-j_*- • •*y q.)- = x i * - • •* x p*y- L*- • - * y q f o r every 46. p > o , q > o , x±,y±. e V . The a d m i s s i b i l i t y of the sequence £G m j 3Sn 3m>l 1 S e q u i v a l e n t to the e x i s t e n c e of c e r t a i n b i l i n e a r maps, i . e . , p r o c e e d i n g as i n the above example of the Grassmann a l g e b r a , we d e f i n e f o r every p >_ ° , q >_ o , the r e l a t i o n s B c ( V(P> x V < * > ) x V ( p + q ) as B p , q = .*x ,y 1*...*y ) , 3 C j * . . . *x * y 1 » . . . *y q)} f o r every p > o , q > o , and B ^ , B ^ and B „ as the J c ' ^ J 0,0 p,o o,q obvious b i l i n e a r maps. Then, the sequence f G rn i , Xm 3m>l 1 S a d m i s s i b l e i f and o n l y i f B i s a b i l i n e a r map f o r p, q every p >_ o ,; q >_ o . As B p Q (resp. B Q q ) i s a b i l i n e a r map f o r every p >_ o (resp. q >_ o) , we need o n l y concern o u r s e l v e s w i t h the r e l a t i o n s B , p > o , q > o . A c r i t e r i a f o r the b i l i n e a r i t y of the r e l a t i o n B i s g i v e n i n the f o l l o w i n g way: p, q For each ( y i a . . . ,y ) e , d e f i n e B p ( y 1 , . . . , y q ) : V p - V ( p + q ) by ( B p ( y i , . . . , y q ) ) ( X l , . . . ,x p) = X j * . . . * x p * v i * - • • * v q > and f o r each ( x 1 , . . . , x p ) e V^ p^ , d e f i n e B q ( x 1 , . . . , x p ) : V q - V ( p + q ) by ( B q ( X l , . . . , x p ) ) (Y1>--->YQ) = x!*-• •*x p*y 1*- . . * y q . Then 5.28 P r o p o s i t i o n : A necessary c o n d i t i o n f o r B to be ir J Vt b i l i n e a r map i s t h a t every B p(y- L,. . . ,y q) i s a (G p,X p)-map and every B q ( x ] L , . . . ,x ) i s a (G q,X q)-map. 47 . Proof: Suppose that f o r each f i x e d (x-^,...,x p) e V p , the map B q(x-,,. . . ,x ) i s a (G , X )-map . Hence there e x i s t s a unique l i n e a r map B^(x^,. . . ,x p ) : V^- 1 - y ( p + q ) such that B 4 ( x p . . . 3 x ) i s a commutative diagram, i . e . , ( B q ( x , ,. . . ,x )) (y., *.. . *y ) x-j^*. . . *Xp*y^*... * y q (see 3 - 2 7 ) . Consider the map : V P - H o m ( v ( q ) , V ( p + q ) ) g i v e n by ( X l,...,x p) - B q ( x 1 , . . . , x p ) T h i s map i s a (G ,X )-map, t h i s b e i n g immediate from the f a c t that B ( y 1 , . . . , y ) i s a (G ,X_)-map, f o r every p 1 q P Jr (y - j L,...,y ) e V^ - . Hence, there e x i s t s a unique l i n e a r map B : V^- 1 - Hom(V^q^, V ^ p + q ^ ) such that B ( x 1 * . . . * x p ) = B q(x^ ,. . . ,x ) , i . e . , we have a b i l i n e a r map V^- 1 x -» ¥(p+q) g i v e n by (x-^. . . *x .y.^. . . *y ) - x^*.. . * x p * y 1 * . • • *y which i s j u s t B p > q Q.E.D. Example: L e t H be a t r a n s l a t i o n - i n v a r i a n t group (not n e c e s s a r i l y reduced), i . e . , H ' - 1 - ' < H , and l e t X : H -» F* be a c h a r a c t e r f o r H . We say th a t the p a i r (H,X) i s t r a n s l a t i o n - i n v a r i a n t i f X(a^ 1-') = X(a) f o r every a e H . Given a t - i p a i r (H,X) , l e t H N denote, as b e f o r e , [a e H / a(m) = m f o r every m > n} , f o r each 4 8 . n >_ 1 . L e t X n = X / - the r e s t r i c t i o n o f X to . Then, we say t h a t £ H n . > x n 3 n>i i s t h e a s s o c i a t e d sequence of  the p a i r (H,X) . We note that i f a e H p , then a e H p + q and X p + q ( a ) = X_(a) f o r every q >_ o . Furthermore, e H and X ^ f a ^ ) = X (a) . We have the f o l l o w i n g : P+q p+q P 3.29 P r o p o s i t i o n : The a s s o c i a t e d sequence of a t r a n s l a t i o n -i n v a r i a n t p a i r i s a d m i s s i b l e f o r every v e c t o r space V . Proof: L e t {H ,X 1 v n be the a s s o c i a t e d sequence of a L n' n J n > l ^ t - i p a i r (H,X) . L e t V be a f i x e d , but a r b i t r a r y v e c t o r space, and f o r every n >_ 1 , l e t V^ n^ be a f i x e d ( H n J X n ) - s p a c e over V . L e t B p q P > o , q > o be the r e l a t i o n s d e f i n e d b e f o r e . We must show that every B_. i s a b i l i n e a r map. By 3-28, i t s u f f i c e s to show that p > q every B p ( y 1 , . . . , y q ) i s an (H p,X p)-map and every Bq(x-, ,. . . ,x ) i s an (H ,X )r-map f o r every p > o , q > o . J . p q q Consider any B ( y . , . . . , y ) : v p - V ^ p + q^ . L e t o e ^ . Then, a e H p + q and ( B p ( y ^ . ., , y q ) ) ( X Q ( 1 } , . . . ,KQ ( P } ) = x a ( l ) * . . . * x a ( p ) * y i » . . . * y q = X P + q ( a ) U l * ' - ' * V y l * - - ^ y q ) = X p ( a ) ( x 1 * . . . * x p * y i * . . . * y q ) = X p ( a ) ( B p ( y 1 , . . . , y q ) ) ( x 1 , . . . , x p ) . i . e . , B p ( y 1 , . . . , y q ) i s an (H p,Xp)-map . Consider any B q ( x 1 , . . . , x p ) : V q - v ^ p + q ^ . L e t T e H q . Then € H p + q and 49. ( B q ( x 1 , . . . , X p ) ) ^ T ( i r . . . * y T ( q ) ) = x 1 * . . . * - K p * y T ( 1 ) * . . . * y T ( q ) = Vq ( T C P 3 ) ( xl*'"*Vyl*---# yq ) = X ( T ) ( X 1 * . . .*x p*y ]_*. . .*y ) - = X q ( T ) ( B q ( x 1 , . . . J x p ) ) ( y 1 , . . . , y q ) i . e . , B q ( x ^ , . . . , x p ) i s an (H q,X^)-map. Hence, every B p q i s a b i l i n e a r map and ^ - ^ ^ n ^ r ^ i a d m i s s i b l e f o r V , where V was chosen a r b i t r a r i l y . Q. E.D. 2. S t r o n g l y A d m i s s i b l e sequences. We have shown i n the p r e v i o u s s e c t i o n t h a t the a s s o c i a t e d sequence t ^ n ^ ^ n ^ i o f a t - i P a i r ( H s x ) i s a d m i s s i b l e f o r e v e r y v e c t o r space V , i . e . , we can c o n s t r u c t an a l g e b r a o f the 'symmetry c l a s s e s o f tensors a s s o c i a t e d w i t h H^ and X n over "V ' . We s h a l l now c l a s s i f y the sequences which are a d m i s s i b l e f o r every V and show that the t r a n s l a t i o n - i n v a r i a n t p a i r s a r i s e n a t u r a l l y from such sequences. By a p a i r (H,X) we mean H < S^ and X i s a c h a r a c t e r of H . Then (H 1,X 1) < (H 2,X 2) i f < H g and X 2 / E± = X± . By (H^X-^ = (Hg,X 2) we mean ( H l j X l ) < ( H2^ X2^ 9 X 1 6 H i = H 2 • Consider a p a i r (H,X) . Define, f o r every p > o , x'-P"1 : H t p ] - F* by X C p ] ( a ^ ) = X(a) where 50. a e H . Then, X L^ J i s a c h a r a c t e r of H L^ J and the p a i r ( H ^ ^ X ^ 3 ) i s denoted by ( H , X ) ^ . Now, c o n s i d e r a sequence of p a i r s { H n , X n } n > j , where < S n f o r every n >_ 1 . Then, fHn'Vn^l i S s a i d to be s t r o n g l y - a d m i s s i b l e ( s - a d m i s s i b l e ) i f i t i s a d m i s s i b l e f o r every v e c t o r space V . Before we proceed to d i s c u s s the s - a d m i s s i b l e sequences we need the f o l l o w i n g : Remark: L e t V be an i n f i n i t e - d i m e n s i o n a l v e c t o r space. Then, every ( H n , X n ) - s p a c e V^ n^ i s n o n - t r i v i a l . For, l e t £ ei]j_ €i b e a b a s i s f o r V , and l e t be the space spanned by e^,...,e n . I t w i l l be shown i n s e c t i o n 4,. t h a t V^ n^ 4= o . In f a c t , the v e c t o r ,e^*...*e n 4: ° > a n d i f some a e S n i s such that a 1 , then e^*...*e n and e a ( l )*" * ' * e a ( n ) a r e l i n e a r l y _ : 3 - n d e P e n d e n t i n v| n^ . Now, as we have an onto map V -• , i t induces an onto map y ( n ) _ y ^ ) m Hence, V^ n^ 4= ° 3 X 1 ( 1 furthermore, e 1*...#e n and e a ( i ) * « • • * e a ( n ) a r e 1 i n e a r l y ~ i n d e p e n d e n t i n v"( n) f o r every a e S such that a 1 H n . 5.50 P r o p o s i t i o n : £**n*^n^n>l i s s - a d m i s s i b l e i f and o n l y i f f o r every n >_ 1 /> ( H n ' x n ' < 'Hi+l'Vt-l' 5 1 . Proof: Suppose £ - ^ X n ^ n > l s a t i s f i e s c o n d i t i o n (*<) . I t f o l l o w s t h a t (Hp.Xp) < ( H p + q , X p + q ) and ( H q , X q ) [ p ] < ( H p + q , X p + ( i ) f o r every p,q > o . Then, p r o v i n g f H ^ X j . } ^ a d m i s s i b l e f o r every V i s analogous to the p r o o f of 3 . 2 9 . Conversly, suppose {-^ Xn^n>l i s s ^ ^ i s s i b l 6 • Let V be i n f i n i t e - d i m e n s i o n a l w i t h b a s i s {e-^,e^,e^,,. . .} . As remarked above, every ( H n , X n ) - s p a c e over V" , - V^n^ i s n o n - t r i v i a l . Consider v^ n +^' 1 f o r any f i x e d n >_ 1 . Suppose there e x i s t s a e such that a <J H n + ^ . Then, by above remark, the v e c t o r s e C T ( i ) * • • • * e a ( n + i ) a n c ^ e-^*. . . * e n 4 j _ a r e l i n e a r l y - i n d e p e n d e n t . But by a d m i s s i b i l i t y e /-,\*...*e / n \ = (e / n\*...*e / \)*e , = a ( l ) a(n+1) v a ( l ) a ( n ) y n+1 (•^n^0^ e l * * * ' * en^* en+l = \ ( a ^ e l * ' *' * e n + l ) > ±' e-> ea(l)*''**ea(n+l) ~ ^n^a)ei*-•**en+l = ° 5 which c o n t r a d i c t s l i n e a r - i n d e p e n d e n c e . Hence, H^ < ^ and then *n+l/\ * s X n ' . 1 ' 6 - (V^) < (WVl ) a n d s i m i l a r l y , (H n,X n)'- 1-' < ( H ^ - ^ X ^ ) f o r every n >_ 1 . Q. E.D. Example: Consider the sequence ^ H n ^ x n ^ r i > l w n e r e % ~ H 2 == 1 , H 5 = A 5 = { 1 , ( 1 2 3 ) , ( 1 3 2 ) } • Set X 5 ( ( 1 2 3 ) ) = ?J1 ( p r i m i t i v e cube r o o t of u n i t y ) . F i n a l l y , s e t = S n and X n = e = 'sign of the permutation' f o r every n _> 4 . L e t V have dimension 3 over a f i e l d F , where V l e F . 5 2 . Then, i f CharP 4= 2 , we have } = A(V) =.o f o r every n > 4 . Then, B i s t r i v i a l l y b i l i n e a r when p+q > 3 > P y q and as (H-^X-J < (H 2,X 2) < (H^,X^) , B p q i s b i l i n e a r f or p+q _< 3 , i . e . , {Hn-'Xn^n>l i s A D M I S S I N L E F O R V J B U T i s not s-admissible. Remark: Let {^n^^n-Wl ^ e 9 X 1 s~a'^m^ss^-1°^-e sequence. Then, new s-admissible sequences may be constructed from {H n,X n3 n > 1 as follows: Let p >_ o and define r B£ = 1 , X^ = 1 l < n < p . [ H n + P = > X n + P - X n P ] f o r e v e r y n > 1 ' i . e . , ^^n J^n^n>l 1 S t 3 i e s e ( l u e n c e ^^n j Xn^n>l w h e r e each EL^  permutes the numbers p+1, p+2,. . ., p+n . The s-a d m i s s i b i l i t y of t-^'--^} follows immediately from that of f H n , X n 5 n > l : Another sequence may be constructed as follows: Let q >_ o and define = 1 ' ^  s 1 1 £ n <_ q . H' = H X' = X f o r every n > 1 . n+q n n+q n J — i . e . , { ^ X ^ j ^ i i s t h e sequence CH n,X n} n > 1 'shifted' upwards, where each E^ i s considered a subgroup of & n + q • The s - a d m i s s i b i l i t y of t ^ n ^ ^ n ^ ^ i i s again immediate. F i n a l l y , we can combine the above constructions 53-i n the f o l l o w i n g way: Given a p a i r (p,q) where p _> o , q >_ o d e f i n e 1 ^ = 1 , = 1 1 < n < p+q ^ p + q = H n P ] > Xn+p+q = ^ f 0 r e V e ^ n > 1 Then, the s-admissibleesequence ^ ^n^ x n^n>l l s ca^-'-e<-i t l i e ( p , q ) - t r a n s l a t e of { S n > x n } n > 1 • The n o t i o n of an s- a d m i s s i b l e sequence b e i n g a (p,q)-t r a n s l a t e of another sequence l e t s us r e s t r i c t our a t t e n t i o n to s - a d m i s s i b l e sequences which are 'reduced' i n the f o l l o w -i n g sense: An s - a d m i s s i b l e sequence {^n* Xn^n>l "*"s s a :"- d t o be reduced i f e i t h e r i t i s the t r i v i a l sequence (HL^ = 1 f o r every n >_ l ) or i f f o r some m > 1 , moves the i n t e g e r s 1 and m . I f moves 1 and m , and as < Hm+-^ and H^1'' < H m + ] _ » i t f o l l o w s that H m + 1 moves 1 and m+1 , i . e . , moves 1 and n f o r every n >_ m . We now show t h a t every s - a d m i s s i b l e sequence i s a ( p , q ) - t r a n s l a t e of a uniquely-determined reduced s - a d m i s s i b l e sequence f o r some p _> o , q >_ o . Before proceeding w i t h p r o o f we note t h a t by (H-L,X1) = (H 2vX_J ^" p^ we mean that (Hg,X 2) = 54. 5.31 P r o p o s i t i o n : Every s t r o n g l y - a d m i s s i b l e sequence i s a ( p , q ) - t r a n s l a t e o f a u n i q u e l y determined reduced sequence. Proof: L e t £ E ^ x n } b e a n o n - t r i v i a l s - a d m i s s i b l e sequence and suppose t h a t i s i t s f i r s t n o n - t r i v i a l group, N > 1 , For every n >. N , l e t r n denote the g r e a t e s t i n t e g e r moved by . Then, 1 < r n <_ n , and as H^ 1 3 < V l ' t h e n V l m o v e s r n + 1 . 1- e-> r n + l > r n f o r every n >_ N . Set q n = n - r n f o r every n >_ N . Then, q n >. >. %x+2 !.>'''> hence e v e n t u a l l y , q n = q f o r some q >_ o f o r every n >_ m , where m >_ N . L e t p be the l a r g e s t i n t e g e r such that a l l the groups H^ f i x the i n t e g e r s l , 2 , . . . , p . Then p >_ o . We show t h a t ^ H n ^ x n ^ n > i i s t h e ( p , q ) - t r a n s l a t e of a reduced sequence ^^ n^ n^-Wl " Now, as i s n o n - t r i v i a l , then N = r N + q ^ >_ T-§+(l > p+q , hence d e f i n e ^ H n j X n ^ n > l a s f o l l o w s : = 1 < n < N - ( p +q) ( \ > \ ) = ( ^ ( p + q J ^ + C p - K i ) ^ ^ 3 f o r every n _> N-(p+q) Then, t^n^n-Wl i s c l e a r l v s - a d m i s s i b l e sequence whose f i r s t n o n - t r i v i a l group i s % _ ( p + q ) = ( ^ ) ^ ~ P 3 • Furthermore, as EL^ moves the i n t e g e r (p+l) f o r some n >_ N , then ^ . ( .p+q) = ( H n ) ^ " P " ' moves the i n t e g e r 1 . 55. S i m i l a r l y , moves the i n t e g e r (n-q) f o r some n >_ m , and then H ^ p ^ moves the i n t e g e r n-(p+q) . i . e . , E v e n t u a l l y , every moves the i n t e g e r s 1 and n , hence £ H n j' Xn^n>l i s a r e d u c e d s-£dmissible sequence. By i t s c o n s t r u c t i o n , i t i s u n i q u e l y determined Q.E.D. L e t { H ^ X ^ } ^ be a ( p , q ) - t r a n s l a t e o f ^ H - r i ^ n ^ l • f ,° r s o m e P > o , q > o . We s h a l l i n d i c a t e the c o n n e c t i o n between the corresponding a l g e b r a s o f 'symmetry c l a s s e s of t e n s o r s ' . L e t (resp. V ^ ) be a f i x e d ( H n , X n ) -space (resp . ( H ^ X ' )-space) over V . L e t A(V) = E V^ n^ n 1 1 n> o and A(V) = £ 7 ^ . n>_o Now, i f i s the f i r s t n o n - t r i v i a l group, then N > p+q , i . e . , = 1 , 1 <_ n _< p+q , hence VKn/ S ® V 1 <_ n <_ p+q . Consider , where n > p+q . Then, f i x e s the i n t e g e r s l , 2 , . . . , p and n-q+l,...,n . On the remaining n-(p+q) i n t e g e r s , i t s a c t i o n i s isomorphic t o f r i + a ) • I n such a case, i t has been shown (see [ 7 ] , 4 .1 ) t h a t f [ n ) = (®V) ® v^ n"" p~ q ' ®(®V) (vector-space isomorphism), f o r every n > p+q . Furthermore, i f n = p+q , 56. -i \ p+q. P Q. P i~\ P then V ( l P + q ; ® V * ( ® V ) ® F ® ( ® V ) = (®V)®V^ o ;®(®V) so the isomorphism i s extended f o r a l l n>_ p+q . Then, A ( V ) = s v < n ) = ; s v ( n ) 0 s v( n> n>o o<n<p+q n>p+q ( £ ® V ) © ( 2 (®V) ® V ^ n " p " q ^ ® ( ® V ) o<n<p+q n>p+q n p q _i ( 2 ® V ) @ ( ( ® V ) ® A ( V ) ® (®V)) . cKn<p+q For the remainder of t h i s p r e s e n t a t i o n , by an s - a d m i s s i b l e sequence and a t r a n s l a t i o n - i n v a r i a n t group, we s h a l l always mean a reduced one. L e t ^-^n-»Xn--n>l ^ e a n s ~ a ( ^ i n i s s i t ) l e sequence. Then, as B^ < H n + 1 f o r every n >_ 1 , we can d e f i n e H = U < S . Furthermore, f o r every a e H , d e f i n e n> 1 n X(a) = X j o ) i f a e . Claim: X i s w e l l - d e f i n e d and (H,X) i s a t - i p a i r . By s - a d m i s s i b i l i t y , = x m f o r every o < m <_ n and f o r every n >_ 1 , hence X i s w e l l - d e f i n e d . Now, i f a e B then a e , f o r some m >_ 1 , and a'-1-' e H^1-' < < H , i . e . , < H . Furthermore, X(o^) = X m + 1 ( a [ l ] ) = X ^ l ] ( a C l ] ) = X m ( a ) -= X(a) , i . e , (H,X) i s a t - i p a i r . 57. Then, there i s a n a t u r a l way of corresponding a t r a n s l a t i o n - i n v a r i a n t p a i r w i t h an s- a d m i s s i b l e sequence. T h i s enables us to d e f i n e an equivalence r e l a t i o n on the s- a d m i s s i b l e sequences; i . e . , two s- a d m i s s i b l e sequences are s a i d to be e q u i v a l e n t i f t h e i r c o r r e s p o n d i n g t - i p a i r s are the same. Given a t - i p a i r (H,X), we l e t [H,X] denote the e q u i v a l e n c e c l a s s of s- a d m i s s i b l e sequences whose c o r r e s ~ ponding t - i p a i r i s (H,X) . L e t *VVn>l > K>^n>l € [E>^ ' Then we d e f i n e the r e l a t i o n < on [H,X] by f ^ j X ^ } < ^ X n } n > l i f 911(1 o n l y i f ( W < f o r e v e r y n >_ 1 . Then, {[H,X]^<] i s c l e a r l y a p a r t i a l l y ordered s e t . Furthermore, i f t Hn.; , Xn^n>l 1 S t h e a s s o c i a t e d sequence of the t - i p a i r (H,X) , then £ ^ x n } n > i < *Wn>l f o r e V e r y {l^XnW € [H>X] > * Wn>l 1 8 t h e m a x i m a l element of the p a r t i a l l y - o r d e r e d s e t . I t i s i n t h i s sense t h a t the a s s o c i a t e d sequence of the t - i p a i r (H,X) serve as the bes t r e p r e s e n t a t i v e f o r the e q u i v a l e n c e c l a s s [H,X] , i . e , , i t re p r e s e n t s the ' f u l l e s t ' s - a d m i s s i b l e sequence i n [H,X] . We conclude t h i s s e c t i o n by showing that e q u i v a l e n t sequences are equal, except f o r the f i r s t f i n i t e number of terms i n the sequences. L e t {H ,X } v_ be the a s s o c i a t e d sequence of a L n' n J n > l ^ t - i p a i r (H,X) and suppose ^^>-^)ny±. € t H * x ] • N o w > 58. i n 1.3, we have shown t h a t i f H i s of type-k , k >_ 1 , then e v e n t u a l l y , the o r b i t s f o r the a c t i o n of on {l , 2,...,n} are Z. = {i,i+k,i + 2 k , . . . } n{l , 2 ,...,n) , i , n 1 _< i _< k , and H^ c o n t a i n s a l l even permutations on Z. „ , i . e . , f o r some m > 1 , A_- x . . . xA. < H f o r x,n __,n K,n n every n >_ m . In the pr o o f o f 1.3 we u t i l i z e d o n l y the f a c t s t h a t there e x i s t s an M such t h a t f o r every n >_ M , moves the i n t e g e r s n l and n , and t h a t En < H Hp""' < H n + 1 f o r every n >_ 1 . Now, as the groups H^ 's s a t i s f y these c o n d i t i o n s , I t f o l l o w s t h a t f o r some m' > 1 , A-j_ n x - • • xA^ . n < H^ f o r every n >_ m' . Hence: 3.32 ..Proposition: ^n , Xr_--n>l e '•^jX-' l f o n l y ^ f o r some M >_ 1 , (H^,X^) = (^,2^) f o r every n >_ M . Proof: C l e a r l y a s u f f i c i e n t c o n d i t i o n . Suppose then that {H^,X^} n > 1 e [H,X] . L e t { H n , X n 3 n > 1 be the a s s o c i a t e d sequence and l e t 3 be the r e p r e s e n t a t i o n map as i n 1.6, i . e . , 3(H)'••= C where C i s a c l o s e d subgroup of P(k) , f o r some k >_ 1 . Now, i f a e H^ , then a e H and 3(a) i s de f i n e d . L e t C n = <*(H^) , Then, C n i s a subgroup (not n e c e s s a r i l y c l o s e d ) o f C , f o r every n >_ 1 . As H^ < E'2 <. . . < H = U H^ we have ^ < C 2 <....< C == U C n . Then, f o r some m" >_ 1 , we have C n = C f o r every n >_ m" . A l s o , there e x i s t s m' >_ 1 such t h a t A 1 n x . .. xA^ n < H^ 59-f o r a l l n > m' . L e t M = max(m',m") . Now, H/ < H — v 3 3 n n f o r every n >_ 1 , hence f o r every n >_ M , C = 3 5 ( H ^ ) < 3 f ( H n ) < a(.H) = C , i . e . , 3(H^) = C = 3 ( 1 ^ } . Then, f o r every a e t h e r e e x i s t s T e H^ 3(a) = 3 ( T ) , i . e . , O T " 1 e Ker3 = A o o(k) . But, as Q T " 1 i s i n , then i t must be i n A-. x...xA, „ . A l s o , as n >_ M _> m' , so X jj XI K. y XX Al,nX—-^ Sc^ n < K and aT_1 € K ' B u t a s T e H n 113 f o l l o w s t h a t a e H^ , i . e . , H n < ^ f o r every n _> M . I t f o l l o w s that = f o r every n _> M and so (K>K) = f o r e v e r y n > M • Q . E . D . We remark that i f and { ^ X ^ ) n > 1 are e q u i v a l e n t s - a d m i s s i b l e sequences, then t h e i r c o r r e s -ponding a l g e b r a s o f 'symmetry c l a s s e s of t e n s o r s ' are not n e c e s s a r i l y the same, i . e . , l e t V^ n^ ( r e s p . V^ n^) be an ( H n , X n ) - s p a c e over V (resp. (H^,X^)-space over V ) , and l e t A = E v( n ) and A = S V^ n^ . Then, as the n>o n>_o sequences are e q u i v a l e n t , we know that V^ n^ = V^ n^ f o r every n _> M , f o r some M _> 1 . However, A need not be isomorphic to A 3- A l g e b r a s of symmetry-classes of t e n s o r s . We have shown t h a t g i v e n a t - i p a i r (H,X), we can c o n s t r u c t an a l g e b r a of 'symmetry c l a s s e s o f t e n s o r s ' f o r each a d m i s s i b l e sequence i n [H,X] . I t i s , however, Go. the a l g e b r a c o n s t r u c t e d f o r the a s s o c i a t e d sequence o f (H,X) which i s the most n a t u r a l g e n e r a l i z a t i o n of the c l a s s i c a l Tensor, Grassmann and Symmetric algebras,, i . e . , i t can be d e f i n e d by a s c e r t a i n u n i v e r s a l mapping p r o p e r t y . Before p r o c e e d i n g w i t h our d e f i n i t i o n , we need some p r e l i m i n -a r i e s : L e t (H,X) be a g i v e n t - i p a i r . Then, we have shown (see 1.9) t h a t H i s completely determined by a unique permutation oQ , the generator of H , and i s denoted by H = [aQ] . L e t N be the l a r g e s t i n t e g e r which i s moved by a . I f a = 1 , we l e t N = 1 . o o Le t {H ,X } v , be the a s s o c i a t e d sequence of n' nJn>_l ^ (H,X) . Then, a Q e , where N i s minimal. Le t V be a v e c t o r space and A an a l g e b r a ( a s s o c i a t i v e w i t h u n i t ) over the f i e l d F , and l e t N f : V -» A be a l i n e a r map such that TT fix /. \) = . N i = l a o ( l ) X(a ) TT f (x. ) , x. e V . D e f i n e f o r every n >_ 1 , ^ i 1 """ n f n : V n - A by f n ( x - _ , . . . ,x n) = > 7r f(x±) . Then: 3.3J> Lemma: For every n >_ 1 , f i s an (H^X^.-map . Proof: As OQ e H^ . , and N i s minimal, then H n = 1 f o r every 1 < n < N , i . e . , f R i s t r i v i a l l y an ( H n , X n ) -map f o r 1 < n < N . 6 1 . Now, i f a e H , n > N , and f (x /,v,...,x , \) = . 3 n 3 — 3 . n v a ( l ) 3 a ( n ) y X ( a ) f n ( x 1 , . . . , x n ) f o r every x. e V , then we say that f n i s symmetric w i t h r e s p e c t to a . In p a r t i c u l a r , we are giv e n that f N i s symmetric w i t h r e s p e c t to aQ . I t then f o l l o w s immediately from the d e f i n i t i o n o f f n that i f a , T e and i f f i s symmetric w i t h r e s p e c t to a and T , then f n i s symmetric w i t h r e s p e c t to ar and f n + 1 i s symmetric w i t h r e s p e c t t o a and . As aQ generates , i t f o l l o w s that f ^ i s an (H^jX^O-map. Assume that f I s an (H n,X n)-map f o r some n _> N . Then, by the above remarks, i f a,T e , then f n + 2 . x s symmetric w i t h r e s p e c t to O T ^ 3 . But, by 1 . 9 , every permutation i n ^ ^ Is a product o f permutation o f the form aT^ "1"-' , a,T e , i . e . , ^n+i i-s a n ( H n + l ^ X n + l ) - m a p • By i n d u c t i o n , f i s an (H^X^)-map f o r every n >. 1 • Q.E.D. L e t (H,X) be a t r a n s l a t i o n - i n v a r i a n t p a i r , where' H = [a] and a moves N , where N i s maximum. Le t V be a v e c t o r - s p a c e , A an a l g e b r a and l e t f : V -» A be a l i n e a r map such t h a t TT f ( x /.\) = X(a) TT f ( x . ) , i = l ° { x ) i=I x x^ € V . Then: 62. 5.54 D e f i n i t i o n : We say t h a t A ( t o g e t h e r w i t h the map f ) i s an, a l g e b r a of symmetry c l a s s e s of tensors w i t h r e s p e c t t o [H»X] over V i f f o r every l i n e a r map g of V i n t o an a l g e b r a B such t h a t TT g(x = X(a) TT g(.x. ) i = l a u ; i = l 1 t h e r e e x i s t s a unique homomorphism cp : A -» B such t h a t g = cp-f , i . e . , such that f V > A B i s commutative. I f A e x i s t s , we say, simply, t h a t A i s an r [H , X ] a l g e b r a over V . 5.55 P r o p o s i t i o n : Given a t r a n s l a t i o n - I n v a r i a n t p a i r ( H , X ) , then an [ H , X ] - a l g e b r a over V e x i s t s and i s unique up to isomorphism (of a l g e b r a s ) . Proof: L e t £ H n / x n 3 n > i b e t h e a s s o c i a t i v e sequence of the p a i r ( H , X ) , and l e t A = s , where = P , m>o V^ 1^ = V , and V^ m) i s an ( H m , X m ) - s p a c e over V , f o r every m _> 1 . Then we have seen (5-29) t h a t A i s an a l g e b r a . L e t f : V A be g i v e n by f (x) = x . Then, . N N ^ f ( x a ( i ) ) = x o ( l ) * . . . * x a ( N ) = X N ( a ) ( x 1 * . . . * x N ) = X ^ T T ^ X . ) We c l a i m t h a t A ( t o g e t h e r w i t h f ) i s an [ H , X ] - a l g e b r a over V . 6 3 . L e t g : V - B , B an a l g e b r a , be l i n e a r and N x . . N , . such that F g ( x / , \ ) = X(a) TT g(x. ) . Then, d e f i n i n g i = l • i = l m 1 g m : v™ - B by g m ( x ] [ , . . . ,x m) = TT g(x±) , we know ( 3 - 3 3 ) that g_. i s an (H m,X m)-map f o r every m >_ 1 . Hence, f o r every m >_ 1 , t h e r e e x i s t s a unique l i n e a r cpm : - B such t h a t cp^x-.*. . . #xftl) = fi^C^. • • •. x^ ) = ? s(x ±). L e t cpQ : V^ 0- 1 -» B be gi v e n by cp 0(a) = ct-Lg ', and l e t cp : A -» B be the unique l i n e a r map which extends a l l the cp ' s . Then cp(l.) = 1_ and, furthermore, l e t t i n g x = x 1 * . . . * x p , y = y 1 * . . . * y q , p > o , q > o , then cp(x-y) = cp(x 1*. . . *x p*y- L*. . . * y q ) = cp (x1*. . . *x^*yJ*. . . *: = xr g(x. )• ? g ( y j = cp n(x,*. . .*x )-cpn(y-|*. . .*y_) = cp(x)cp(y) . i = l i = l " " " -^But as the v e c t o r s o f the form x,y generate A (as a v e c t o r - s p a c e ) , i t f o l l o w s t h a t cp : A -» B i s a homomorphism. F i n a l l y , t o show uniqueness,of cp , suppose t h a t cp' : B - A i s such that cp'-f = g . Then, cp = cp' on a l l elements of the form f ( x ) , x e V . But, these elements are c l e a r l y a s e t of a l g e b r a i c generators f o r A . Hence, cp = cp Q.E.D. I f A i s an [H,X]-algebra over V , then A i s denoted by V[H,X] . 64. Examples: L e t H = 1 , X = 1 $ then V [ l , l ] = ® V - the t e n s o r a l g e b r a . L e t H = , X = e = ' s i g n of the permutation 13 then V [ S ,e] = A ( V ) - the Grassmann a l g e b r a (CharF ^ 2) . In e i t h e r o f the above ' c l a s s i c a l ' cases. D e f i n -i t i o n 3-34 reduces to the well-known d e f i n i t i o n s of the t e n s o r and Grassmann a l g e b r a s , as g i v e n i n ([3],[4]). A remark on the c h a r a c t e r X of a t - i p a i r ( H , X ) : L e t (H,X) be a t - i p a i r . Then, X ( a ^ 1 3 ) = X ( a ) f o r every a e H . L e t a Q be the generator of H , then by 1.9, every T e H has the form a ^ 1 ! 3 . . . a^ 1* 1 3 , and hence X i s completely determined by X(a ) , and X ( T ) i s e i t h e r 1 or -1 f o r every T e H . X can be one of the f o l l o w i n g c h a r a c t e r s : (a) X = 1 - the t r i v i a l c h a r a c t e r . (b) X = e - the 'sign o f the permutation' c h a r a c t e r . As a and a ^ ± 3 have the same p a r i t y , -(o^^) = e(a) for, every a e H . We note that i f the generator aQ i s an even permutation, then e reduces to the t r i v i a l c h a r a c t e r . (c) L e t 3 be the r e p r e s e n t a t i o n map, and suppose t h a t 3(H) = C , 3 ( a Q ) = a 0 , i . e . , C = [ a Q ] . F u r t h e r -more, suppose that C i s non-minimal In P(k) . Then we have remarked a f t e r 2. l4, t h a t we can d e f i n e 65. 1 i f P has an even r e p r e s e n t a t i o n i n C . -1 i f p has an odd r e p r e s e n t a t i o n . 1*1 1 C l e a r l y , p(P ) = l-i(P) f o r every p e C , and we can ' l i f t ' u to a c h a r a c t e r ( a l s o denoted by \x) of H , such t h a t ia(o )•= -1 . We note that i f a Q i s an odd permutation, then u reduces to e . ^- B a s i s f o r V[H,X] as a vec t o r - s p a c e . Suppose (H,X) i s a t - i p a i r where H = [ a Q ] and x ( c 0 ) =~1 • ket v be f i n i t e dimensional. In c o n c l u d i n g t h i s p r e s e n t a t i o n we s h a l l show t h a t V[H,X] i s a l s o f i n i t e dimensional as a ve c t o r - s p a c e . We make the f o l l o w i n g p r e l i m i n a r y remarks: L e t G < S m and X : G - F* a c h a r a c t e r o f G . Suppose t h a t V i s f i n i t e - d i m e n s i o n a l over F , w i t h b a s i s y-j_, • • • ,y n- I t was shown (see [6 ] , [7]) that the (G,X)-space over V , V^(G) i s f i n i t e - d i m e n s i o n a l . In f a c t , l e t S = set of a l l sequences w = (w^ ,...*"w ) w i t h 1 < w. £ n and d e f i n e an a c t i o n of G on S by w -» w' where wCT = ( w a - l ( i • • • > w a - l ( m ) ) w € S , a e G . The group a c t i o n s p l i t s S i n t o d i s j o i n t o r b i t s , so l e t R' be a set of r e p r e s e n t a t i v e s f o r these o r b i t s . F o r each w = (Wl,...,w ) e R' we l e t y w = y w * - . . * y w e v£(G) . 1 m .a 66. L e t (G) denote the s t a b i l i z e r of w , i . e . , 'w ( ;G) W = ia e G / wa = w} . L e t R c R' be d e f i n e d as [w e R' / X(a) =1 f o r every a € (G) 1 . Then (see [ 7 ] ) the v e c t o r s y . w e R , form a b a s i s f o r V^(G) . I f R = 0 then V^(G) = (o) . In1 p a r t i c u l a r , i f dim V = m , then V^(G) ^ (o) . For c o n s i d e r w = (1,2,...,m) . Then w can be taken as a r e p r e s e n t a t i v e o f the o r b i t c o n t a i n i n g w , i . e . , w e R' But c l e a r l y , wG = w i f f a = 1 hence X(a) = 1 , i . e . , w e R and y T r = y 1 * . . . * y m =f  o . Furthermore, i f r e S w ± m m and T 4 G , then w' = ( T ( l ) , . . . , T ( m . ) ) i s a d i f f e r e n t o r b i t than w and as (G) / = 1 , w' e R' , i . e . , y^*-• ••*ym and * * * ^ T(m) a r e ^ n ^ e P e n ^ e n ^ v e c t o r s i n v £ ( G ) . L e t (H,X) be t - i and c o n s i d e r V[H,X] where V i s f i n i t e - d i m e n s i o n a l . Now,, i f X = 1 , then (H,X) < ( S ^ l ) . From D e f i n i t i o n 3-34, i t f o l l o w s t h a t t h e r e e x i s t s an onto homo-morphism V[H,X] - . But as V t S ^ l ] = S(V) - the symmetric a l g e b r a which i s known to have i n f i n i t e dimension, so V[H,X] i s a l s o an i n f i n i t e - d i m e n s i o n a l v e c t o r - s p a c e . Suppose then, t h a t H = [a ] and X ( a Q ) = -1 (CharF ^ 2) . Suppose a l s o , that the type o f H i s k , k > 1 , i . e . , A (k) < H < S (k) . We note that H + A (k) as X 4= 1 • Then (by 1 . 9 ) , oQ i s the permutation 6 7 . ( i ^ i ^+k). . . ( i ^ i^+k) where 1 = i 1 < ± 2 < . . . < i t <_ k . L e t N ;be the l a r g e s t i n t e g e r moved by a , i . e . , N = l t + k . L e t y-|_j...,y n be a b a s i s f o r V , and c o n s i d e r V[H,X] = S V^ m^ , where f o r every m >_ 1 , V^ m^ i s an m>_o ( H m , X m ) - s p a c e over V . We c l a i m t h a t V[H,X] i s a f i n i t e -d imensional v e c t o r - s p a c e . In f a c t , set N* = N + ( n - l ) k , then: 5 . 5 6 P r o p o s i t i o n : V^ m^ = (o) f o r every m _> N* and N* i s minimal. Proof: We have i n d i c a t e d that f o r every m >_ 1 , V^ m^ has f i n i t e b a s i s y T r *...*y T r where 1 < w. < n and the sequence W - , w__ — 1 — 1 m w = (wj, ...3w ) e Fv^ , where R m i s as d e f i n e d above. 1 We want to show t h a t = 0 f o r every m >_ N* . L e t Z. = { i , i+k, l+2k, . . . } n{l,2,...,m} ,1 < i < k , f o r J_ j III "* ~~ every m _> 1 . L e t w = ( W j ,. . . ,wm) and 1 _< w^  <_ n . Then, f o r a f i x e d m':> 1 , d e f i n e the set w. = {w. / j e Z. } ( c o u n t i n g - X J X j i l l r e p e t i t i o n s ) f o r every 1 <. i <_ k . Now, i f m >_ N* = N + ( n - l ) k = ( i t + k ) + ( n - l ) = i t + n k , then the s e t s wo ,...,w. have each at l e a s t (n+l)-elements. ^ l 1 t But, as 1< w^  <_ n f o r every 1<_ i _< m , i t f o l l o w s t h a t i n each of the sets w. , 1 < j < t , a t l e a s t two 68. elements have the same v a l u e , say: w. + r .k = w. + s .k , where r . ={= s . , J J j = 1 , 2 , . . . , t . Now, l e t T be the permutation (i^+r^k,. i^+s^k) . . . ( i t + r t k I t + s t k ) . , then, as aQ = ( i ^ i-^+k) ... ( ± t i t + k ) i t f o l l o w s t h a t a Q T i s a product o f even permutations, i . e . a T e A- x...xA. (where A. _ denotes the a l t e r -o l,m K,m v i,m n a t i n g group on Z. ) and hence, by Theorem 1 . 3 , J. y III • ° o T e \ 3 ± ' e " T e \ ' Then, X ( C T 0 T ) = 1 > a s x ( a 0 ) = > w e n a v e X ( T ) = - 1 . Hence, we have wT = w , i . e . , T e ( \ l ) v r > and X ( T ) = - 1 . Th e r e f o r e , w Rffi , and as w was chosen a r b i t r a r i l y , we must have R^ = 0 , f o r every m >_ N* We complete the p r o o f by showing N* to be minimal . L e t m = N * - l and c o n s i d e r w = (w1,...,wm) . Le t w^  be d e f i n e d as b e f o r e , 1 <_ i £ k . Then as m < N* = I t+nk , i t f o l l o w s that the set s w\ , w^  + 1 , . . . ,w, have o n l y n-elements i n each. In p a r t i c u l a r we can choose w f o r which w^ . = {l , 2,...,n} f o r j = i t , i t + l , . . . ,k . Suppose t h a t f o r some T e ^ , wT = w , i . e . , T e ( H m ) w . Now, T = T 2 - - - T k where each permutes the l e t t e r s i n Z. , 1 < i < k . Then, T. must permute the l e t t e r s i n x,m 3 — — 3 x " _ x f o r 1 <_ i <_ k , and as w = w , we c l e a r l y must have T j = 1 , f o r j = i t , i t + l , . . . , k . I f T j y . . . ^ then 3>(T) 4= 0 "(where 3 i s the r e p r e s e n t a t i o n map) k 6 9 . and ! | 3 ( T ) | | < i = ||g? (a ) || , which c o n t r a d i c t s the hypothesis t h a t oQ i s the generator. T h e r e f o r e , T e A-, x...xiL and then, X ( T ) = 1 . Hence, f o r a JL j Til _j III s p e c i a l c h o i c e of w , we have X ( T ) L 1 f o r every T e ( E ^ ) ^ i . e . , the v e c t o r y ' = y #...*y i s not zero, and ' W W - . w VM _ v(m-i) + ( 0 ) _ 1 Q.E.D. Example: L e t ( H , X ) .= (5^,6) , then V [ H , X ] i s the Grassmann a l g e b r a , which we know i s f i n i t e - d i m e n s i o n a l , when V i s . Now, H = [ a Q ] , where aQ i s the t r a n s -p o s i t i o n ( 1 2 ) , i . e . , a - moves the i n t e g e r N = 2 , and H i s o f type-k , where k = 1 . Then, i f V i s n-dimensional then N* = N + ( n ^ l ) k = n+l , and we get the well-known r e s u l t t h a t A ( V ) = o f o r every m > n and A ( V ) =f= ° • BIBLIOGRAPHY [ l ] N. Bourbaki, Algebre M u l t i l i n e ' a i r e , Herman, P a r i s , 1958, Chap, 3-[2] ¥. Burnsid e , The Theory of Groups, Cambridge Univ. Press, 1911. [3] C. C h e v a l l e y , Fundamental Concepts of A l g e b r a , Academic P r e s s , N.Y., 1956-[4] C. C h e v a l l e y , The C o n s t r u c t i o n and Study of C e r t a i n  Important A l g e b r a s , Tokyo, Math. Soc. of Japan, 1955- . [53 W. Greub, M u l t i l i n e a r A l g e b r a , Academic P r e s s , N.Y., 1967 • [63 M. Marcus and H. Mine, Permutation on Symmetry C l a s s e s . Jour, of A l g e b r a , 5, (1967) , pp. 59 -71 . [7 3 K. Singh, On the V a n i s h i n g of a Pure Product i n a  (G,q) Space, Ph.D. T h e s i s , U.B.C., 1967• [83 H. Wielandt, F i n i t e Permutation Groups, Academic Pr e s s , N.Y., 1964. [93 M. H a l l , Theory of Groups, Macmillan Co., N.Y., 1959-

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