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The group ring for S₃ Botta, Earle Peter 1963

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THE GROUP RING FOR by E a r l e Peter B o t t a B„A., The U n i v e r s i t y of B r i t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA # f A p r i l , 1963 In presenting 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 requirements.for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I f u r t h e r agree- that per--mission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying, or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed . without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia,. Vancouver 8, Canada. Date • r\pvJi: ~3>o \ 1 b3, 11 A b s t r a c t The u n i t s i n the group r i n g f o r over the i n t e g e r s are i n v e s t i g a t e d I t i s shown t h a t the o n l y u n i t s of f i n i t e o r d e r are of o r d e r two, t h r e e o r s i x - I n f i n i t e c l a s s e s of u n i t s o f each of t h e s e o r d e r s are c o n s t r u c t e d as w e l l as an i n f i n i t e c l a s s of u n i t s of i n f i n i t e o r d e r , T The e q u a t i o n G = AA , where G i s a uniraodular group m a t r i x of r a t i o n a l i n t e g e r s and A a m a t r i x of r a t i o n a l i n t e g e r s , i s i n v e s t i g a t e d i n the r i n g of group m a t r i c e s f o r S^. I t i s shown t h a t A = CP, where C i s a unimodular group m a t r i x of r a t i o n a l i n t e g e r s and P a g e n e r a l i z e d p e r m u t a t i o n m a t r i x . I t i s a l s o shown t h a t i f H i s a p o s i t i v e d e f i n i t e symmetric u n i -T modular group m a t r i x then H = H^H^ where i s a group m a t r i x o f r a t i o n a l i n t e g e r s and H i s of i n f i n i t e o r d e r except i n the t r i v i a l case when H = I« I hereby c e r t i f y t h a t t h i s a b s t r a c t s a t i s f a c t o r y , i v Acknowledgment The author \irould l i k e to thank h i s s u p e r v i s o r . Dr. R« C» Thompson, f o r h i s p a t i e n t a s s i s t a n c e i n the p r e p a r a t i o n of t h i s t h e s i s . He would also l i k e to thank the N a t i o n a l Research C o u n c i l of Canada f o r t h e i r f i n a n c i a l support. i i i T a ble of Contents Page 1« Group Rings 1 2» The l e f t r e g u l a r r e p r e s e n t a t i o n of a f i n i t e group 2 3. U n i t s i n a group r i n g 4 4» The e x i s t e n c e of n o n - t r i v i a l u n i t s i n a group r i n g 6 5« The group r i n g f o r 7 6. U n i t s i n the group r i n g f o r 9 T 7• The e q u a t i o n G = AA i n the r i n g of group m a t r i c e s f o r 20 8, B i b l i o g r a p h y 35 1 1« Group Rings Let G be any m u l t i p l i c a t i v e group and Z the r i n g of r a t i o n a l i n t e g e r s * The set of a l l f i n i t e formal sums n g.. x f Z, g. c G w i l l be denoted by Z(G)I . l g . i i = l g i 1 g i Z ( G ) can be made i n t o a r i n g by d e f i n i n g a d d i t i o n (+) and m u l t i p l i c a t i o n .(•) as follows,. n n Suppose x, y ^ 2(G); x = ^ _ x g , y = ^ y i = l g i 1 j = l g - j g j n n then (a) x«y = ^ (x y )g, g., t ~ T g • g • 1 J i = l j = l y i J n (b) x+y = ^ U + y^ )g. . x g, g. l 1 = 1 l i -It i s not hard to v e r i f y that {z(G-)# +, »} i s an a s s o c i a t i v e r i n g with i d e n t i t y l e , where e i s the i d e n t i t y i n G« D e f i n i t i o n . (z(G), +, •} i s c a l l e d the group r i n g  f o r G over Z or simply the group r i n g f o r G« In what f o l l o w s the i d e n t i t y matrix w i l l be denoted by I. The phrase reif and only i f t s . w i l l be a b b r e v i a t e d to «iff«. 2 2» The l e f t r e g u l a r r e p r e s e n t a t l o r t of .a f i n i t e group Let G be a f i n i t e group of o r d e r n and suppose the elements of G are (g^# g R ) i n some f i x e d o r d e r . F o r each g £ G c o n s i d e r the o r d e r e d s e t (g g,, •»•, g g )o s s J. s n T h i s set i s some p e r m u t a t i o n of ( g ^ f • »•», g^) so (g gg 1# g g g n ) = (g x# •••# g a ) p ( g g ) where P ( g B ) i s a p e r m u t a t i o n m a t r i x a s s o c i a t e d w i t h g . The ( i , j ) element of P ( g ) i s 1 i f g. = g g. and i s 0 o t h e r w i s e . De-*s l s j f i n e the symbol g s U , J ) = f l i f g.g" 1 = g g 0^ o t h e r w i s e then P ( g s ) = ^ g ^ 1 ' j ) ) a n d p ( g s ) p ( g t ) = p ( g s ^ t ^ s i n c e n 5 1 g s ( i # r ) g t ( r # j ) = g s g t ( i , j ) , f o r g g ( i # r ) = 1 i f f r = l g r = g g ^ i a n d g t ( r # j ) = i i f f g r = g t g j so n / g ( i , r ) g ( r , j ) = 1 i f g g, = g.g,"" and 0 o t h e r w i s e . , S S • S "t 1 J r = l I t i s c l e a r t h a t P ( g i ) = P(g.j) i f f g± = g.' D e f i n e a map f : G _ M^(Z), where ^n^^ i s ^he r i n g of n-square m a t r i c e s over Z, by f ( g ^ ) = P ( g ^ ) . I t i s c l e a r from the above d i s c u s s i o n t h a t f i s an isomorphism. D e f i n i t i o n . The s e t of n-square p e r m u t a t i o n m a t r i c e s P ( g i ) # i = n, g^ £ G i s c a l l e d a l e f t r e g u l a r r e p r e s e n t a t i o n of G in ( Z ) • 3 The m a t r i c e s P(g^) are l i n e a r l y independent s i n c e g (i# j ) = g+Ci/ j ) = 1 f o r some i f j i m p l i e s -1 g, = g.g. = g • * t i j s Let x = x g, £ Z ( G ) and d e f i n e a map t = l g t t n F: Z ( G ) -. M ( Z ) by F(x) = / x P ( g , ) . Then F(x) = (x - l ) n J. i 9J. t g.g. t = l * t i j s i n c e the matrices P(g^.) do not have non-zero elements i n eommon p o s i t i o n s . Since the m a t r i c e s P(g^.) are l i n e a r l y independent F(x) = 0 i f f x - 0 so the map i s 1 - 1* F u r t h e r F(x • y j - (x -1 + y - l ) = U - l ) + (y„ - l ) gi<si g ^ . . g ^ = F(x) + F(y), # and F(xy) = F ( x ) F ( y ) s i n c e F ^ g ) = P C g ^ j ) = P ( g i ) P ( g j ) = F ( g i ) F ( g j ) . Hence F i s a r i n g isomorphism of Z ( G ) i n t o M N ( Z ) and the image of F i s the set of m a t r i c e s of the form (x -l)« g.g. D e f i n i t i o n . Let £ Z ( G ) » The matrix t = l g t * X = (x - l ) £ M ( Z ) i s c a l l e d the group m a t r i x f o r x» g i g j n 4 U n i t s i n a. group r i n g D e f i n i t i o n B Let G be a group and Z ( G ) the group r i n g f o r G. An element x £ Z ( G ) i s a l e f t ( r i g h t ) u n i t i f f t h e r e e x i s t s a y £ Z ( G ) such t h a t xy = l e (yx = l e ) where l e i s the i d e n t i t y i n Z(G)o An element x £ Z ( G ) i s a u n i t i f f i t i s both a l e f t and r i g h t u n i t . D e f i n i t i o n . Let X be an n-square m a t r i x . X i s unimodular i f f det X = + l a Theorem 1. Let G be a f i n i t e group. I f x £ Z ( G ) then x i s a u n i t i f f the group m a t r i x f o r x i s u n i m o d u l a r B P r o o f . Suppose x i s a u n i t i n Z ( G ) . Then t h e r e e x i s t s a y £ Z ( G ) such t h a t xy = l e . Let X and Y be the group m a t r i c e s f o r x and y r e s p e c t i v e l y 8 Then XY ~ I so det XY * det X«det Y = 1. Hence det X = det Y = + 1 s i n c e det X, det Y are r a t i o n a l i n t e g e r s . C o n v e r s e l y , suppose t h a t X i s the group m a t r i x f o r an element x £ Z ( G ) and X i s u n i m o d u l a r . Let n n n n element of Z ( G ) . Then where So 5 (I) /z \ g l . = X / g i Take z = 1 i f g, i s the i d e n t i t y and z = G otherwise* Since X i s a matrix of r a t i o n a l i n t e g e r s the above system of equations ( i ) can be s o l v e d f o r y . • « » , y i n i n t e g e r s . v l y n Then xy = l e £ Z ( G ) 0 Let Y be the group m a t r i x f o r y. Then XY « I so Y = X" 1 and s i n c e X X - 1 = X _ 1 X = YX = I i t f o l l o w s that yx = l e . Hence x i s both a l e f t and r i g h t u n i t . The above proof can be found i n If G i s a f i n i t e group then every l e f t ( r i g h t ) u n i t i s a l s o a r i g h t ( l e f t ) u n i t . Suppose x i s a l e f t u n i t . Then there e x i s t s a y such that xy = l e . Let X and Y be the group matrices f o r x and y r e s p e c t i v e l y . Then XY = X X - 1 = X _ 1 X = YX = I so yx = l e and x i s a r i g h t u n i t , If G i s any f i n i t e group then the set of u n i t s i n Z ( G ) form a m u l t i p l i c a t i v e group 0 Suppose x and y are u n i t s . Then t h e r e e x i s t x ^, y ^ such that x ^x = xx ^ = l e -1 -1 ~ -1 -1 -1 -1 , and y y = y y = l e so y x xy = xyy x and xy i s a u n i t . 6 4» The e x i s t e n c e of n o n - t r i v i a l u n i t s i n j | group r i n g D e f i n i t i o n , Let G be any group and Z ( G ) the group r i n g f o r G« A u n i t x £ Z ( G ) i s t r i v i a l i f i t i s of the form + l g f o r some g £ G. I f x i s not of t h i s form i t i s n o n - t r i v i a l . D e f i n i t i o n . I f x £ Z ( G ) i s a u n i t then x i s of f i n i t e o r d e r i f f x° = l»e f o r - some p o s i t i v e i n t e g e r n. I f n i s the l e a s t such i n t e g e r x i s s a i d t o have o r d e r ,n. I f no such i n t e g e r n e x i s t s x i s s a i d t o be of i n f i n i t e o r d e r . I f G i s a f i n i t e group the q u e s t i o n of the e x i s t e n c e of n o n - t r i v i a l u n i t s i n Z ( G ) has been c o m p l e t e l y s o l v e d . Higman £l] proves t h e f o l l o w i n g theorem. Theorem. I f a l l elements of a group G have f i n i t e o r d e r , Z ( G ) has n o n - t r i v i a l u n i t s u n l e s s G i s e i t h e r ( i ) an A b e l i a n group the o r d e r s of whose elements a l l d i v i d e f o u r or ( i i ) an A b e l i a n group t h e o r d e r s of whose elements a l l d i v i d e s i x or ( i i i ) t he d i r e c t p r o d uct of a q u a t e r n i o n group and an A b e l i a n group, t h e o r d e r s of whose elements a l l d i v i d e two. In t h e s e cases Z ( G ) has o n l y t r i v i a l u n i t s . 7 The group r i n g f o r Let be the symmetric group on three symbols and Z(S„) the group r i n g f o r S„• If the elements of S_ are J i 3 g l = g 2 = ( 1 2 3 > ' g 3 = ^ 3 2 ) , g 4 = ( 1 2 ) , g 5 = (13) and g^ = (23) then the group m a t r i x X = ( x g g-t) f o r an element x x g, € 2(S_) i s ( l e t t i n g x i = l g i 1 J . 9 : / X l X 3 X 2 x 4 x 5 x 6 \ x 2 x l x 3 x 6 x 4 x5 X = x 3 x 2 X l x 5 x 6 X 4 x 4 x 6 X5 x l x 2 x 3 I'5 x 4 x 6 x 3 x 4 x 2 / Xr-•> x 4 x 2 x 3 x l ' Suppose th e elements of S 3 o t h e r than ( g 1 # , say ( g r > 1 • 0 e m a t r i x X 1 = (2 g r . g r 1 1)0 j Let P be the with a one i n row i , column r. 1 1 i = 1, P TX»P = X. D e f i n i t i o n . Let A and B l e t C = B) 0 T ^ e n c i s c a l l e d the d i r e c t sum of A and B and we w r i t e C = A * B. 8 Note that X = A B T T B A T T where A, B, A , and B are 3-square c i r c u l a n t s . Let U « CO 1 G G O 0 0 0 W2 1 CO u) l oJ u.2 1 0 0 0 -a -a -a a a a l a a a a a ai wh e r e CO = - i + ,n J UXU Y = -1 = Y * Y * £ x * e 2 where x^ - x 2 + co (x^ - x 2 ) x 4 " x 6 + w 2 ( x 5 " x 6 } Then D i s u n i t a r y and x 4 ~ x 6 + ^ ( x 5 " x 6 }  x l ~ x 2 * w 2 ( x 3 " x 2 } £ 1 = x l x. £ 2 = x l + x 2 x 3 " x 4 " x5 " x 6 x 3 + x 4 + x5 + x 6 " D e f i n i t i o n . Let X be a n-square m a t r i x . The t r a c e  of X, denoted by t r X , i s the sum of the main d i a g o n a l elements of X. Let E 0(x.., x j , Xj,) = x.Xj + x, x,. + x_. x,_« Then det Y = x-' 2 v ~ i 2 . 2 2 . 2 . 2 + x 2 + x^ - (x^ + X(- + x^) - E 2 ( x 1 # x 2, x^) • E 2 ( x 4 , x 5 , x 6 ) t r Y = 2x^ - x 2 - x^» Since x^ ( i = 1 # •«•, 6) i s a r a t i o n a l i n t e g e r , det Y, t r Y, £^ and £,> are r a t i o n a l i n t e g e r s , 9 6« U n i t s i n the group r i n g f o r Theorem 2. The only u n i t s of f i n i t e order i n Z(S^) are of order two, three or s i x . P r o o f 0 By theorem 1, to determine the u n i t s i n Z(S^) i t i s s u f f i c i e n t to determine the unimodular group m a t r i c e s f o r S-« If X = (x . L ) i s a group m a t r i x f o r g i g j x g Z(S^) then X i s s i m i l a r to Y * Y * &1 * £ 2 # where Y, £^ and £,, a r e a s ^ n S e c t i o n 5. If det Y = + 1, e- = + 1, e 2 o + 1 then s i n c e det X = (det Y ) 2 ^ ^ , X i s unimodular. Conversely, i f X i s unimodular then det Y = ± 1, e i = + 1, & 2 = + 1 s i n c e det Y, e and £ 2 are r a t i o n a l i n t e g e r s . Since X i s s i m i l a r to Y * Y * e * £ 2 , X n = I i f f Y n « I, £* = 1 and £ 2 = 1. Lemma 1. If det X = + 1 then E 2 ( x 1 # x 2, x^) = E2^ x4» x 5 ' x 6 ^ where E 0 ( x . , x., x,) = x.x. + x.x, + x , x, • 2 l ' j k i j l k j k P r o o f . det X = + 1 i f f det Y = + 1, £^ = + 1 2 2 2 2 2 2 and £ 2 = + 1. + 1 = det Y = x^ + x 2 + - x^ - x^ - x^ - E 2 ( x 1 # x 2 , x 3 ) • E 2(x^,Xj-,X£) + 1 = e 1 £ 2 = x^ + x 2 + x^ - xjr - x^ - x? + 2 E 2 ( x 1 , x 2» x^) 2 E 2 (x^# Xj- , X^ ) a So £ 1 £ 2 - det Y = 3 [ E 2 ( x 1 # x 2 r x ) - E 2 ( x ^ , x $ , x 6 ) ] = G, + 2. 10 S i n c e E^Cx^, x^p x^) and E^{x^, x^, x^) are r a t i o n a l i n t e g e r s the only s o l u t i o n i s E ^ x ^ x^# x^) = E ^ x ^ , x^, x ^ ) . Lemma 2 a Let X be unimodular with i n t e g r a l e n t r i e s . Then Y « c l i f f X = c l . Proof» Suppose Y = c l . Since Y has a l g e b r a i c i n t e g e r s as elements c i s an a l g e b r a i c i n t e g e r . Since t r Y = 2x^ - x 2 - x^ = 2c i s r a t i o n a l , c i s a r a t i o n a l 2 i n t e g e r 8 Then det Y = + 1 = c i m p l i e s c = + Is. The c o n d i t i o n Y = c l i m p l i e s x„ = x 0 , x. = xc = xLt and <• i 4 ~> o x'^  - x 2 = c. Since X i s unimodular £^ = x 1 + x2 * - x^ - x^ . - x^ = • 1 and e 2 - x± + x 2 + x 3 • x 4 + x 5 + x 6 - + 1. £ 1 + E 2 Hence x^ + x 2 + x^ » 2" 6 (» 0 , +. l ) £ - £ 2 1 and x^ + x,j + x^ = 2 — ( «= 0 , • l ) B Since x^ - x 2 = c EX * e 2 and x 2 = x^, 2 ; - c = x.^  + x 2 + x^ - (x^ - x^) = 3 x 2 > £ 1 *' £ 2 But v 1 " - c = 0 , + 2. Hence x_ = 0 , Since x. = x r = X/. 2 2 4 5 o x, + x c + x, = 3x, «= £ 2 ~ £ 1 . But £ 2 "" £ 1 = 0 , i 1. Hence 4 5 6 4 2 2 x^ = 0» Thus x 2 = x^ = x^ = x^ = x^ = 0 . Then x^ = = e 2 and 0 = 2 - c> Hence x^ = c and X = c l 0 If X = c l then, s i n c e X i s a matrix of r a t i o n a l i n t e g e r s , i f X i s unimodular o E + 1, Since U X t f 1 = c l = Y * Y * e 1 + e , Y = c l . 11 Lemma 3a Let m(A,) be the minimal polynomial f o r Y, Then m(A,) i s a monic polynomial with r a t i o n a l i n t e g e r s as c o e f f i c i e n t s and i s of degree one or two 0 If m(A.) i s of degree two i t i s the c h a r a c t e r i s t i c polynomial f o r Y 0 Proof• If m(?0 i s l i n e a r then Y = c l so by lemma 2 c = + 1 and m(A.) = A. + 1* If m(?0 i s of degree two then s i n c e i t i s monic 2 and d i v i d e s the c h a r a c t e r i s t i c polynomial A. - ( t r Y)A. + det Y of Y # m("A.) = A.2 - ( t r Y)A, + det Y. Therefore s i n c e t r Y and det Y are r a t i o n a l i n t e g e r s m(A.) has r a t i o n a l i n t e g e r c o e f f i c i e n t s . S i nce m(A.) d i v i d e s the c h a r a c t e r i s t i c polynomial the degree of m(A.) cannot be g r e a t e r than two. Lemma J^a Suppose x g Z(S^) s a t i s f i e s x P = l e , where p i s a prime g r e a t e r than t h r e e . Then x = l e . Pro o f , Let X be the group ma t r i x f o r x. Then . X P = I, Let m(?0 be the minimal polynomial f o r Y, By lemma 3 m(7v.) i s a monic polynomial with r a t i o n a l i n t e g e r c o e f f i c i e n t s of degree one or two. Case ( i ) : m(A.) i s l i n e a r . Then Y = c l so by lemma 1 X = c l and c = + 1, If c = -1 then X P = - I c o n t r a d i c t i n g X P = I, Hence X = I and x = l e . Case ( i i ) : m(A.) i s of degree two. As X P = I, Y P = I« Hence A.P - 1 i s an a n n i h i l a t i n g polynomial f o r Y 12 and m(A,) d i v i d e s A.P - 1 0 The unique f a c t o r i z a t i o n of A,P - 1 over the r a t i o n a l number f i e l d i n t o i r r e d u c i b l e f a c t o r s i s [53s TJ3 - 1 = (K - l ) ^ ' 1 + ••• . + A. + 1 ) . Hence m(A,) = (A. - l ) (A,P * »•<> + X + l ) where e^ and e^ are 0 or 1. If p > 3 there i s no choice of exponents e^ # e 2 that makes deg m(A.) two. k ~ e. Suppose x £ Z(S«) i s of order n and n = | I p . 1 3 - i = l 1 (e. > 0) i s the c a n o n i c a l f a c t o r i z a t i o n of n i n t o prime 1 k e . e . p J power f a c t o r s a Let m = | I p . 1 then ( x m ) J = x 1 1 = le« i = l 1 Hence i f . x n = l e and p|n f o r some prime P > 3 then x m = l e where m = —•» Hence i f a u n i t of order n e x i s t s P then n * 2 1 3 j . 2 i 3 j Lemma 5« Suppose x ^ = l e , i > 2. Then 2 i - 1 3 j - l e . x P r o o f . Let x« = x * . Then (x») = l e . Let X be the group matrix f o r x* and m(A.) the minimal p o l y -nomial f o r the a s s o c i a t e d Y. Then deg m(A.) i s one or two. Case ( i ) : m(A.) i s l i n e a r . Then Y = c l so by 2 2 lemma 2 X = c l and c = i 1. Hence X = 1 and ( x f ) = l e . 2 2 i " 1 3 j Since ( x 1 ) = x t h i s i m p l i e s the r e s u l t -Case ( i i ) : m(?0 i s of degree two. Since X^ " = I, Y^ " = I so - 1 i s an a n n i h i l a t i n g polynomial f o r Y« The 13 unique f a c t o r i z a t i o n of ?\A - 1 i n t o f a c t o r s i r r e d u c i b l e over the r a t i o n a l number f i e l d i s A/4" - 1 = (A, - 1) (A. + l ) (A. +' l)» 2 By lemma 3 m(A.) = A, - ( t r Y)A. + det Y has r a t i o n a l c o e f f i -c i e n t s so we have two p o s s i b i l i t i e s (a) m(A.) = A, - 1 (b) ' m(?v.) = A,2 + 1. Case ( a ) : If m(A.) = A,2 - 1 then Y 2 = I so Y 2 * Y 2 + • e 2 » I. Hence X 2 = I so ( x » ) 2 = ^ = l e . 2 Case (b) : If m(A.) = A, +1 then s i n c e ra(Aj = A,2 - ( t r Y)A. + det Y i t f o l l o w s that t r Y = 2x± - x 2 - x^ = 0 and det Y = 1. Since t r X = hx± = 2 t r Y + + e 2 ; t r Y = 0, e± = + 1, e 2 = + 1 i m p l i e s t r X = 0 so x^ = 0 and = - £ 2 « Since e l = x l + x 2 . * . x 3 " x4 ~ x5 " x6 a n d e 2 = x x + x 2 • x 3 + x 4 • x 5 * x 6 # e x + e 2 = 0 = x± * x2 * x3 and x^ + x^ + x^ = • 1» Hence x 2 = -x^» Using lemma 1 and the above r e s u l t s g i v e s (1) det Y = 2x2 - ( x 2 • x 2 + x|) = 1 (2) E 2 ^ x i * x2' x3^ ~ ~x2 = X4 X5 + x4 xo + X5 X6 = E2^ x4' x5' x6^° M u l t i p l y i n g equation (2) by two and adding i t to equation ( l ) g i v e s - [ ( x 2 + x 2 «• x 2 ) * 2(x 4x 5 • x 4 x 6 • x 5 x 6 ) ] - - [ ( x 4 • x 5 • x 6 ) 2 ] = 1 2 but t h i s i s a c o n t r a d i c t i o n s i n c e - (x, + x~ + X/) < 0, Hence 4 ? o — case (b) cannot occur and the proof i s complete. 14 2 i3 j Lemma 6. Suppose x J = l e , j > 2, then 2 13 J- 1 , x = l e . P r o o f . L et x' = x 2 ^ . Then (x») 9 = x 2 3 J l e ' 9 9 Let X be t h e group m a t r i x f o r x f . Then X = I so Y = I . Let. m(A.) be the m i n i m a l p o l y n o m i a l f o r Y a S i n c e A,9 - 1 i s an a n n i h i l a t i n g p o l y n o m i a l f o r Y, m(A.) d i v i d e s A,9 - 1. 9 The unique f a c t o r i z a t i o n of A. - 1 i n t o f a c t o r s i r r e d u c i b l e o v e r the r a t i o n a l number f i e l d i s [5] ?v.9 - 1 = (A, - 1)(A.2 + A, + l ) ( A . 6 + A 3 + 1). S i n c e by lemma 3 m(A,) i s a monic p o l y n o m i a l w i t h r a t i o n a l i n t e g e r c o e f f i c i e n t s of degree one o r two i t f o l l o w s t h a t 2 ' m(A.) = A. - 1 o r m(A.) = A. + A. + 1. Case ( i ) : m (A.) = A, - !• Then Y = I so by lemma 2 3 2"^  3 ~^ X = I so x f = l e . Hence (x*) = x = l e a Case ( i i ) : m(/\) = A,2 + A, + 1. Then Y 2 + Y + I = 0, (Y - l ) ( Y 2 + Y + I) = 0= Y 3 - I . Hence Y 3 = I a S i nee 9 9 9 X = I , e i = e2 = 1 s o t h a t s i n c e 9 i s odd e-1 = e2 = 1. Hence Y 3 * Y 3 + e 3 + e 3 = X 3 = I . T h e r e f o r e (x») 3 = x 2 ^ ± = 1 Combining lemmas 5 and 6 i t f o l l o w s t h a t i f x £ Z(S^) i s a u n i t of o r d e r n = 2*3J then, i , j = 0 o r 1. Hence the o n l y u n i t s of f i n i t e o r d e r are of o r d e r two, t h r e e o r s i x a We w i l l now p r o c e e d t o f i n d i n f i n i t e l y many u n i t s of each of t h e s e o r d e r s as w e l l as i n f i n i t e l y many u n i t s of i n f i n i t o r d e r . 15 The f o l l o w i n g equations w i l l be u s e f u l i n f u r t h e r . i n v e s t i g a t i o n of u n i t s of f i n i t e o r d e r . Using the same n o t a t i o n as b e f o r e , £1 + S (1) x± * x 2 + x 3 = * (« 0, + 1) £ -. £ (2) x 4 • x 5 + x 6 = 2 '2 1 '•(- 0, ± 1) (3) 6xx = t r X = 2 t r Y + £ 1 + (4) t r Y = 2x^ - x 2 - x^. '2 Suppose x i s a u n i t of order two and X the~group 2 2 matrix f o r x. Then Y = I and m(A.)|A, - 1. Hence 2 m(A.) = A, - 1, A. + 1 or A. - 1„ If m(A.) i s l i n e a r then by lemma 2 X = + I s i n c e Y = ± I. 2 Suppose m (A.) = A, - 1B Then by lemma 3 m(A.) i s 2 the c h a r a c t e r i s t i c p o l y n o m i a l f o r Y, A, - ( t r Y)A. + det Y. Hence t r Y = 0 and from (3) t r X = 0, x1 ~ 0. Since t r Y = 2x^ - x 2 - x^ = G i t f o l l o w s that x 2 = x^. From ( l ) and (2) i t i s c l e a r t h a t x^ + x^ + = + 1. Hence i f 2 X =1 e i t h e r x, =0, x» = k # x„ = -k, x. = m, . x c = n, 1 <• 3 4 i> x^ = + 1 - m - n, where k, m and n are r a t i o n a l i n t e g e r s , or X = - I . Since x i s a u n i t , det X = + 1. by Theorem 1. Hence by lemma 1 E2^» ^' = E2^ m* n* - 1 "* m "" H e n c e 2 2 2 k, m and n must s a t i s f y ( i ) k -m - m n - n + m i n = G . Conversely suppose k, m and n s a t i s f y ( i j o Then i f x^ = 0, x 2 = k, x^ = -k, x^ = m, x,- = n, x^ = + 1 - m - n; E 2 ( x 1 # x 2, x^) = E 2 ( x 4 * x5' X D ) s o 2 2 2 2 2 2 det Y = x^ + x 2 + x^ - x^ - x^ - x^ = -1, 16 t r Y = 2x^ - - x^ = 0 , £ 1 = x i + x2 + x 3 *" xl+ ~ x5 ~ x6 = + & 2 = x^ + x 2 + x^ + x^ + x^ + x^ = + 1« Hence 2 2 det X = (det Y) = ~ 1 a n d ^ ~ 1 i s t h e c h a r a c t e r i s t i c 2 2 polynomial f o r Y. Therefore Y «= I so X = I and x i s of order two. If m = k, n = -k equation ( i ) i s s a t i s f i e d . Hence i n f i n i t e l y many u n i t s of o r d e r two are given by x^ = 0, x 2 = k, x^ = -k, x^ = k, x,- = -k, x^ = + 1, where k i s any r a t i o n a l i n t e g e r . Since the choice of two of x., x c and x^ was a r b i t r a r y , two other i n f i n i t e c l a s s e s of u n i t s of order two are gi v e n by x^ = 0, x 2 = k# X3 = ~"k' X4 = - ^' x5 ^ k » x^ = -k and x^ = 0, x 2 ~ ^* x3 = x 4 = k * x,- = + 1, x 6 = -k. Suppose x i s a u n i t of order three and X i s the group ma t r i x f o r x« Then Y 3 = I and m(A.)|A.3 - 1» Since 3 2 Ar - 1 = (A. - l)(A. + K + l ) t h i s i m p l i e s , u s i n g lemma 3r that m (A.) = A. - 1 or m (A.) = AT + A. + 1. It was shown above that 2 m (A.) = A, - 1 then X = I. Suppose m(Aj = ?v. + A. + 1. Since . deg m(A.) = 2, m(A») i s the c h a r a c t e r i s t i c polynomial f o r Y, A.2 - ( t r Y)A. + det Y. Hence t r Y = -1. Using t h i s t ogether with (3) i t f o l l o w s that x1 = 0 and e 1 = £ 2 e l a F r o m ^ and (2) i t now f o l l o w s that x^ + x 2 + x^ = 1, x^ + x^ + x^ = 0 Hence i f x i s a u n i t of order t h r e e , x^ = 0 , x2 ~ ^* x- = 1 —• k, x, = m, x c = n, X/ = -m - n; where k, m and 3 4 i> o n are r a t i o n a l i n t e g e r s . S i n c e x i s a u n i t det X = ± 1 so by lemma 1 k, m and n must s a t i s f y 17 ( I I ) k ( l - k) + ra2 + mn + n 2 = 0. Conversely suppose k, m and n s a t i s f y ( i l ) . If x^ = 0, x 2 = k, x^ = 1 - k, x^ = m, x<- = n, • x^ •= -tm--.-.n, then EgCx^., x^, x^) = E 2 ( x ^ , Xj. , x^) so det Y = x 2 + x 2 + x 2 - x 2 - x 2 - x 2 = 1, t r Y = 2x^ - x2 ~ x3 = ~ 1 ' e i = x i + x2 * x3 ~" x4 ~ x5 ~ x6 = £2 = x1 + x 2 + x^ * x^ + x^ + x^ = 1. Hence det X = (det Y) e 1 e 2 2 = 1 and A. + A. + 1 i s the c h a r a c t e r i s t i c polynomial f o r Y. Therefore Y 2 + Y + I = 0. Hence (Y - l ) ( Y 2 + Y + K) = Y 3 - 1=0, Y 3 = I so I = Y 3 3r Y 3 + e 3 + e 3 = x 3 and x i s of order three, Suppose x £ Z(S^) i s such that f o r some r a t i o n a l i n t e g e r k, x-^  = 0, x 2 = k, x^ = -k, x^ = k, x^ = -k and x^ = + 1« Then as was shown above x i s a u n i t of order twoa R e c a l l g 2 = (123) € S^* Consider y = xg 2x, y^ = 0, y 2 = -3k2, y 3 = 3k2 +1, Yk = ~3k2 • k, = * 2k and v6 = 3k + ka C l e a r l y y £ l e and y 3 = ( x g 2 x ) 3 *= x g 2 x x g 2 x x g 2 x = l e so y i s a u n i t of order t h r e e . This g i v e s an i n f i n i t e c l a s s of u n i t s of order t h r e e . Using g^ w i l l g i v e another c l a s s as w i l l u s i n g d i f f e r e n t c l a s s e s of u n i t s of order two, This technique f o r o b t a i n i n g u n i t s of order three from u n i t s of o r d e r two i s d i s c u s s e d i n Taussky's paper £6]. Suppose x i s a u n i t of order s i x and X i s the group m a t r i x f o r x. Then X^ = 1, so Y° = I and m(A,)|A,^ - 1. 18 Since A.6 - 1 = (A. - 1) (A.2 + A, + 1) (A, + 1) (A,2 - A, + l ) t h i s 2 2 i m p l i e s m(A,) = A. - 1, A. + 1, A, -1, A, + A.+ 1 or A, - A. + 18 If m(A.) i s l i n e a r then by lemma 2 X = + I 2 so X i s not of order s i x . If m(A.) = A. - 1 then as above •2 X = I and X i s not of order s i x . Suppose 2 m(A.) = A. + A- + 1. Then s i n c e m(A.) i s the c h a r a c t e r i s t i c p olynomial f o r Y , t r Y = -1. Using t h i s t o gether with (3) i t f o l l o w s that x.^  =0, e l = E2 = 1 b S i n c e m(A.) = A.2 + X + 1 i t f o l l o w s that (Y - I ) ( Y 2 + Y + I) = Y 3 - 1 = 0. Hence X 3 = Y 3 •* Y 3 + e 3 + e 3 2 = I, a c o n t r a d i c t i o n . Suppose m(A.) = A. - A. + 1. Then s i n c e ra(A.) i s the c h a r a c t e r i s t i c polynomial f o r Y, t r Y = 1. Using t h i s t o g e t h e r with (3) i t f o l l o w s that x.^  =0, e i = E2 = Since (Y + i ) ( Y 2 - Y + I) = Y 3 + I = 0 i t f o l l o w s t h a t X 3 = Y 3 t ? Y 3 * e 3 + e 3 = - I . Hence (-X) 3 = 1 so -X i s a u n i t of order t h r e e . If Z i s a u n i t of order three c l e a r l y -Z i s a u n i t of o r d e r s i x . Hence every u n i t of o r d e r s i x i s of the form -Z where Z i s a u n i t of order t h r e e . There e x i s t i n f i n i t e l y many u n i t s of i n f i n i t e order i n Z(S,j). Suppose x f Z(S^) i s such that x^ = 0, x2 = ^* x^ = -k, x^ = k, Xj. = -k # x^ = + 1 f o r some r a t i o n a l i n t e g e r k. Then x i s of order two. Let X be the group ma t r i x f o r Xo Consider the u n i t y corresponding to the group matrix Y = X TX. If y = ^ i t h e n y± = 4k2 + 1, y 2 = -2k2, y^ •= -2k2, y^ = 2k2 + 2k, y^ = 2k2 + 2k, y^ = -4k2. Since a l l u n i t s y o f f i n i t e o r d e r except + I have y, = 0, y 19 cannot be of f i n i t e o r d e r u n l e s s k = 0» T h i s g i v e s an i n f i n i t e c l a s s of u n i t s of i n f i n i t e o r d e r . 2G 7p The equation G = AA i n the r i n g of group m a t r i c e s f o r S, Let H be any f i n i t e group and suppose i s a unimodular group m a t r i x f o r H 8 If G = AA , where A i s a matrix of r a t i o n a l i n t e g e r s , i s i t p o s s i b l e to f i n d a group T ma t r i x C such that G = CC ? This q u e s t i o n has been answered i n the a f f i r m a t i v e f o r c y c l i c groups by Newman and Taussky [ 4 ] and f o r a b e l i a n groups by Thompson [ 7 ] • This q u e s t i o n w i l l now be i n v e s t i g a t e d f o r the group S^. T Let G = AA be a unimodular group matrix f o r where A i s a m a t r i x of r a t i o n a l i n t e g e r s . As d i s c u s s e d i n s e c t i o n 5 the group m a t r i x depends on the numbering of the elements, of S^, If another numbering of elements i s used T the m a t r i x X i n s e c t i o n 5 i s converted to P XP, P a permuta-t i o n matrix., Since i f D = P T C P , DD T • P T C C T P = P TGP, with-out l o s s of g e n e r a l i t y may be taken i n the form A B \ B T A / T T where A, B, A and B are 3~square c i r c u l a n t s . Let P± = ' P 2 = Il 0 0 0 0 1 \ 0 1 0 Q l - p l 0 P 2 o Def i n i t i o n o The permutation 0" = (12°•«(n-l)n) £ S j i s the n - c v c l e . The matrix P £ M^(z) d e f i n e d by = eij^io»(j) ^ e i j ~ i l ) i s a g e n e r a l i z e d n - c v c l e . 21 P 2 C P 2 • - c T . The f o l l o w i n g lemmas w i l l be needed. Lemma 7» If G i s any 3"~sguare c i r c u l a n t then m Proof. The r e s u l t f o l l o w s by d i r e c t computations Lemma 8. The matrices and Q 2 commute with the m a t r i x G. Proo f . The r e s u l t f o l l o w s by computation, the f a c t that P^ commutes with a l l 3 _ s c P 1 a r e c i r c u l a n t s , and lemma 7« Lemma 9» The matrices A'^-Q^A . and A^Q^A are or t h o g o n a l . P r o o f . For i • 1, 2; ( A ^ Q J i ) ( A ~ 1 Q . A ) T » A ~ 1 Q I A A T Q T ( A " 1 ) T = A A " " 1 A T Q i Q J ' C A 1 ) " 1 « A T ( A T ) ' 1 •« I by lemma 8, Lemma 10. There e x i s t g e n e r a l i z e d permutation m a t r i c e s M 1 and M 2 such that Q^A = A.M^  and -Q^ A = AM 2« Proof. The only orthoganal matrices of r a t i o n a l i n t e g e r s are the g e n e r a l i z e d permutation matrices so by lemma 9 th ere e x i s t g e n e r a l i z e d permutation matrices and such that A - ^ A . - ..M1 and A " 1 Q 2 A « }i m Lemma 11. Let M be a g e n e r a l i z e d permutation m a t r i x . Then M i s s i m i l a r , v i a a permutation matrix, to a d i r e c t sum of g e n e r a l i z e d m-cycles. 22 Proof. The r e s u l t i s o b v i o u s l y t r u e i f M i s a 1-square m a t r i x . Assume the r e s u l t t rue f o r a l l r < n and suppose M i s a n-square g e n e r a l i z e d permutation matrix*. If there i s a non-zero entry i n the ( l , l ) p o s i t i o n of M the r e s u l t f o l l o w s by i n d u c t i o n on the matrix obtained by d e l e t i n g the f i r s t row and f i r s t column of M. If «= 0 then there i s a non-zero element i n the f i r s t row of M. Suppose the non-zero element i s ^ i j " P°st m u l t i p l y i n g M by a permutation matrix P^ interchange the second column and the j*"*V column,, Since l e f t m u l t i p l i c a t i o n of MP^ by P ^ does not a f f e c t the f i r s t row of MP^ P ^ 1 * ^ has a + 1 i n the ( l , 2) p o s i t i o n . If P^MP^ has a + 1 i n the (2, l ) p o s i t i o n the r e s u l t follows- by i n d u c t i o n . If not, then there e x i s t s a + 1 i n p o s i t i o n (2, j ) f o r some j > 3» Interchange columns 3 a ^ d j and rows 3 a n d jo Then e i t h e r the (3# l ) element i s a + 1 i n which case the c y c l e c l o s e s o f f and the r e s u l t f o l l o w s by i n d u c t i o n , or there i s a non-zero element (3* j ) f o r some j > 4« In t h i s case repeat the above p r o c e s s . Since M i s a g e n e r a l i z e d permutation m a t r i x a + 1 must e v e n t u a l l y appear i n column 1. If t h i s happens f o r some i < n the r e s u l t f o l l o w s by i n d u c t i o n . If t h i s happens f o r i «= n M i s s i m i l a r to the n - c y c l e . Lemma 12. Let R be the r i n g of matrices over Z generated by P^ and P 2« Then R = (x e M^CZ): X = x x I + x 2 P 1 + + x^2 +  x^PiP2 + V l P 2 * X i € Z}. 23 2 Proof. C l e a r l y R c o n t a i n s I, P.^ P±t P2, P ^ and ^i^2 ^ P2 P1^° Since these s i x matrices form a represen-t a t i o n of S 0 i n M_(Z) t h i s set i s c l o s e d m u l t i p l i c i t i v e l y . S i nce R i s a r i n g of matrices over Z i t must c o n t a i n a l l l i n e a r combinations-of the above s i x m a t r i c e s . The set •{X-€ M 3 ( Z ) : X = X l I * x 2 P 1 «• x 3 P 2 • x ^ + * x ^ , x i € z} i s a r i n g . Since R i s the sm a l l e s t r i n g c o n t a i n i n g P^ and P 2 i t i s of the d e s i r e d form. Let A. denote the l as a m a t r i x of it's rows: A = th / A x\ A, A, A A, A6> row of A and w r i t i A Then Q A A3 AM 1 = / A l M l \ A2 M1 A3 M1 V l A5 M1 A6M1/ so, s i n c e Q^A = AM^ by lemma 10., A 2 = A^M^, A 3 - A 2M X A, Mf and A c = A.M., A, = A CM 1 = A.M7,. hence 1 1 5 4 1 o ? 1 4 1 ; 24 A = A1 M1 A1 M1 A, A4 M1 •By lemma 11 there e x i s t s a permutation matrix S such that S TM 1S = P * * P where P (j = 1, k) 1 n. n, n . V J ' * . 1 k j i s a g e n e r a l i z e d n. c y c l e . Hence Q,AS = AS (P * • •• & P ) ( J 1 n l n k Since (AS) (AS T) = A S S T A T «= AA T = G we may assume without l o s s of g e n e r a l i t y that = 3- »«• % P^ . S i nee Q 3 = I f ( A " 1 Q 1 A ) 3 = M3 = P 3 P 2 =1 1 k so P 2 . = I f o r a l l j . If n. > 3 P 3 £ I so none of the nj J n j P are g e n e r a l i z e d 4» 5 o.r 6 c y c l e s . If n. = 2 f o r some j then P n C 2 0 and P = + P r £ I. Hence cannot c o n t a i n any 2-cycles« Since n .' = 1 or 3 and n, + + n, = 6 i f M, con t a i n s a j 1 k 1 1-cycle i t must c o n t a i n t h r e e . To show cannot c o n t a i n any l r c y c l e s a technique due to Newman and Taussky i s used [4]« Suppose M^, contains a 1-cycle. Then i t con t a i n s three l-cycles« Two 1-cycles must appear e i t h e r i n the ( l , l ) and (2, 2) p o s i t i o n s or i n the 25 (5, 5) and (6, 6-) p o s i t i onsa Without l o s s of g e n e r a l i t y assume they appear i n the ( l , l ) and (2, 2) p o s i t i o n s . Then (mod 2) has the f o l l o w i n g form. M l * 11 o : \ ' : o o i : o a a a o . a . a o : e-: p (mod 2) where P i s a 4-square permutation m a t r i x . Since A^ = A^M^, A 3 - A ^ , A 5 -f o l l o w i n g form. A s \ M 1 and A 6 • \ M2, A (mod a l l a 1 2 a 1 3 a 1 4 S15 a l 6 \ a l l a 1 2 * « # * 1 a l l a12 * * * * a 4 1 a 4 2 a 4 3 a 4 4 a45 a46 , a 4 1 a 4 2 # * * * / a 4 1 a42 * * * « / (mod 2) The elements i n rows 2 and 3 and columns 3# 6 2 are j u s t ( a ^ , a i 4 * a 1 5 ' a l 6 ^ Permuted by P and P respec-t i v e l y . S i m i l a r l y , the elements i n rows 5 and 6 and columns 2 3, 6 are j u s t (a^3» a 4 4 ' a 4 5 * a 4 6 ^ p e r r a u t e d by P and P r e s p e c t i v e l y a The determinant of A i s now computed modulo two. F i r s t add column 4# 5 and 6 to column 3 0 This leaves det A (mod 2) unchanged and 26 det A = det \ where = a l l a 1 2 c l * * a l l a 1 2 c l # * a l l a 1 2 c l # * a 4 l a 4 2 C 2 * * a 4 1 a 4 2 c 2 * * a 4 l a 4 2 c 2 # # + a 1 4 + a 1 5 + a l 6 (mod 2) c 2 = a 4 3 + a 4 4 + a 4 5 + a 4 6 a l l sums being modulo two. Novr add row one to rows two and three and add row f o u r to rows f i v e and s i x . Then det A (mod 2) i s unchanged and det A s det / a u a 1 2 c l * 0 0 G * # * 0 0 0 * a 4 l S 4 2 C 2 # # 0 0 0 * # * \ 0 0 0 # * */ (mod 2) Columns 1, 2 and 3 are e s s e n t i a l l y 2-vectors over the f i e l d of r e s i d u e c l a s s e s modulo two. Since there are three such v e c t o r s they are l i n e a r l y dependent. Hence m det A = 0 mod 2 . Since G = AA i s unimodular, det A =1 (mod 2 ) , det A = 0 (mod 2) i s a c o n t r a d i c t i o n and cannot c o n t a i n any 1 - c y c l e s . Since cannot c o n t a i n any 1, 2 , 4* 5 or 6 c y c l e s = R^ * where R^ and Rg are g e n e r a l i z e d 3 - c y c l e s . 27 where Let R1 f 0 * 0 o o x2 \X 0 G " * 1» cf X + 1. S i rice I «=. ( A " 1 Q 1 A ) 3 0 cr 0 1 0 0 ^2 cr G 0 3 = f t t T. R 3 or or cr I • t Ot = 1 1 1 2 3 ' 2 1 2 3 ' 1 2 3 3 Let S 1 . 0 0 \ 0 x l 0 0 0 V 2 ' cr, cr or 1 2 3 = 1. 0 0 0 cf1&2 en Then S ^ R ^ = P± and SjRgSg = P][. Let S = S± * Sg, th S ^ S = p ! * p ! = Q]." Hence Q-jAS » ASQ^• Si n c e ( A S ) ( A S ) T « AA T = G without l o s s of g e n e r a l i t y l e t M 2 = Q 1 so Q-jA = A Q 1 # Let A = A l l A 1 2 A 2 1 A22' where the A.. are i j 3-square m a t r i c e s of r a t i o n a l i n t e g e r s . Then Q l A - ( P ^ ^ P ± R 1 2 \ P A PA * l n 2 1 1 22 A 1 1 P 1 A 1 2 P 1 A 1 1 P 1 A 2 2 P 1 ' AQ 1" Hence P,A., 1 i j i ^ j P ^ ( i , j = 1, 2) and s i n c e any m a t r i x that commutes with P, i s a c i r c u l a n t , each of the A., i s a 1 ' • U 3-square c i r c u l a n t . Since A ^ i s a polynomial i n A and the sum and product of c i r c u l a n t s are c i r c u l a n t s . ,-1 when considered 28 as a 2-square m a t r i x with 3-square matrices as elements, has elements that are c i r c u l a n t s . A l s o has r a t i o n a l i n t e g e r elements as det A - ± lm Every 3-square c i r c u l a n t of r a t i o n a l 2 i n t e g e r s i s a l i n e a r combination of I, and P^. Since Q 2 = f 0 P ^ and A~ 1Q 2A = P 2 ° M 2 may be c o n s i d e r e d as a 2-square matrix with elements i n the r i n g R of 3-square m a t r i c e s over the r a t i o n a l i n t e g e r s generated by P^ and P 2 » Let M 2 = / M N M 1 2\ where M. .. - ( i , j = 1, 2) U21 M 2 2 ) i s a 3-square m a t r i x of r a t i o n a l i n t e g e r s . Consider the f i r s t row of M^, Since M 2 i s a g e n e r a l i z e d permutation m a t r i x there i s a + 1 e i t h e r i n or ^\2a Suppose i t i s i n ^ 1 ° ^ th.e non-zero element i s not i n the ( l , l ) p o s i t i o n of M^^ by post m u l t i p l y i n g M 2 2 by a matrix of the form P S- P, where P = P^ or P^, b r i n g the non-zero element to the ( l , l ) p o s i t i o n . Note that s i n c e / M^X I P 0 \ f M 1 X P M 2 1 P \ U21 M22J U P I ~ 1M21P M 2 2 P J post m u l t i p l i c a t i o n by P * P does not s h i f t elements from one block M.. to another- and s i n c e M.. c R, M,,P c R, Since £ R, the r i n g of m a t r i c e s over Z generated by P^ and P 2 # by lemma 12 29 M n P - X l I + x ^ • x 3 P 2 • x 4 P 2 • x 5 P l P 2 + x 6 P 2 P 2  = / x l + \ x 2 + x 6 x 3 + x 5 \ X 0 + X / X , + X C X , + X . 3 o 1 5 1 4 \ x 2 + x 5 x 3 • x 4 x± * x 6/ Since M 2 i s a g e n e r a l i z e d permutation matrix there i s at most one non-zero entry i n each row and column of M^P. S i n c e i t was assumed that x, + x. = + 1 t h i s o b s e r v a t i o n i 4 -r e s u l t s i n the f o l l o w i n g equations: (1) xl + x 4 = + 1 (4) x 2 • x 6 « 0 (2) x 3 + x 6 = 0 (5) x 3 + x 5 = 0 (3) x 2 + x 5 = 0 Equations (2) and (/J.) y i e l d x^ = x 3 and equations (2) and (5) y i e l d x^ = x^« Using these f a c t s has the f o rm M 1 1 P = / x l + x 4 0 0 0 x x + x 5 x 3 + x 4 0 x 3 + x 4 x1 * x 5 If x. + x c = G and x_ + x, == 0 . these equations 1 ? 3 4 together with equation (5) above y i e l d x, + x, = 0 , c o n t r a -1 4 d i e t i n g x^ + x 4 = + 1. Hence **nP = - * 0 1 - ^2 a n C * s i n c e P = I, P1 or P 2; = + I, 3 P±t + p j , + P 2 > 2 - P 1 P 2 o r ~ P 1 P 2 * Since M 2 i s a g e n e r a l i z e d permutation m a t r i x £ 0 i m p l i e s M 1 2 -.^21 ~ 0 s o M 22 i s a 3-square g e n e r a l i z e d permutation matrix« S i m i l a r l y i f M2^ £ 0 30 M l l = M22 = ^ a n d M12 i s a 3-square g e n e r a l i z e d permutation 0 E. ma t r i x . Hence = E± 0 0 E, or M 2 = E, 0 where E^ ( i = 1> 2) i s a 3-square g e n e r a l i z e d permutation m a t r i x . Suppose M 2 = E 1 0 0 E. e By lemma 6 Q 2A = P2 A11 P2A121 PA PA I 2 21 2 22 A11 E1 A12 E2 A21 E1 A22E2, = AM, so A 2 1 = P 2 A n E 1 and A 2 2 = P 2 A 1 2 E 2 . Then A = I 0 0 P 21 A 11 A 12 l A l l E l A12 E2/ Consider det A (mod 2) 0 Since det il 0 \0 P det A = det /A 11 k12 (mod 2)m 'A11E1 A12E2> = l(mod 2) 2/ columns Post m u l t i p l i c a t i o n of A ^ by E^ interchanges the of A ^ i n some way (mod 2), s i n c e E^ i s a permutation matrix modulo 2. S i m i l a r l y post m u l t i p l i c a t i o n of A^ 2 by E 2 interchanges the columns of A ^ 2 i n some way (mod 2). Add columns 2 and 3 to column 1 and columns 5 and 6 to column 4. The determinant of A modulo 2 i s unchanged and 31 det A = det D 1 * * (mod 2) # # n * * 1 # * n * * 1 * * * * * * * *, where denotes the row sum of A ^ and the row sum of .^^ 2" Sinee A ^ and A^ 2 are c i r c u l a n t s the row sums are the same f o r each row of A. and each row of A. . n — — "12* Now add the f i r s t row to each of the ot h e r s to o b t a i n det A = det C 1 * * D 1 * *\ 0 * * 0 * * 0 * * 0 * * 0 * * 0 * * 0 * * o * * \ o * * o * * (mod 2) Columns 1 and 4 are e s s e n t i a l l y two 1-vectors of the f i e l d of i n t e g e r s modulo 2 so are l i n e a r l y dependent and det A = 0 (mod 2) which i s a c o n t r a d i c t i o n . Hence i s of the form / 0 •1] * 2 ° By lemma 10 M 2 = A~ 1Q 2A and s i n c e .Q2 = I ( A ^ A ) 2 - M 2 - ( E±E2 0 = I. 32 —1 T Hence =• E^ = E^# Since Q^A = AQ^ and Q 2A = AM 2 i t f o l l o w s that Q 1Q 2Q 1A = Q 1Q 2AQ 1 = Q i A M 2 Q i = A Q1 M2 Q1" S i n c e Q l Q 2 Q l = Q2 t h i s i m p l i e s Q 2A = AM 2 = AQ^MgQ^ and s i n c e A i s non-s i n g u l a r , Q 1M 2Q 1 - / 0 P ^ N - / 0. E l \ « M 2, V P ^ J P o I \ E J o / Hence P^E^P^ = E^» It has alre a d y been proved that E ^ i s one of + I, + P J # + P 2 , + P 2 , + P ^ g , + P i P 2 ' Since P1 E1 P1 = E l * E l c a n n o t be any of + I, + P^, + P^. Since P1 P2 = P2 P1 a n d P1 P2 = P2 P1 i t f o l l o w s t h a t E l = - P 2 P i (1 < j < 3)» Hence since ( P 2 P : [ ) T = P 2 p i # M2 " -+ ( ° P2 P1 \ \ P 2 P J 0 ' By lemma 6, Q 2 A = / P 2 A 2 1 P 2 A 2 2 \ ± | A 1 2 P 2 P { A U P 2 P J | =AM 2. P2 A11 P2A12 A22 P2 P1 A21 P2 P1 Hence * 2 1 - + & n d A22 " * V l l V l ' R e c a l l A^^ and A^ 2 are 3^-square c i r c u l a n t s so by lemma 7 A21 = - A12 P1 a n d A22 = - A11 P1" S i n c e A i 2 a n d A i i a r e i i T c i r c u l a n t s they commute with P^ so A,^ = + p j A 1 2 and A22 = - P1 A11" Choose k such that j + k = 3, then P ^ + K = I. Let K = / + I ' 0 \ , Then 33 AK = 0 \ A 11 " 1 A11 A l l 12 12 P l A l l 'A A 11 T 12 = C. S i n c e A ^ and A^ 2 are c i r c u l a n t s C i s a group m a t r i x . F u r t h e r C C T = A K K T A T = A A T = G, s i n c e K i s a g e n e r a l i z e d p e r m u t a t i o n m a t r i x . Theorem 3• l e t G be a un i m o d u l a r group m a t r i x f o r T the group S^ and suppose G = AA where A i s a m a t r i x of r a t i o n a l i n t e g e r s . Then t h e r e e x i s t s a group m a t r i x C such t h a t G = CC T. D e f i n i t i o n . Suppose x £ Z(S^) i s a u n i t . Then x i s p o s i t i v e d e f i n i t e symmetric i f f the group m a t r i x f o r x i s p o s i t i v e d e f i n i t e symmetric. T h i s d e f i n i t i o n i s independent of the o r d e r i n which the group elements are t a k e n s i n c e i t was shown i n s e c t i o n 4 t h a t group m a t r i c e s f o r a f i x e d element x £ Z(S„) c o r r e s p o n d i n g 3 to d i f f e r e n t o r d e r i n g s of group elements are s i m i l a r v i a a p e r m u t a t i o n m a t r i x . S i n c e i t i s known [2] t h a t any n-square unimodular p o s i t i v e d e f i n i t e symmetric m a t r i x of r a t i o n a l i n t e g e r s i s of T the form AA i f n < 7 ( t h i s i s f a l s e i f n > 7) the f o l l o w i n g r e s u l t i s a l s o c l e a r . 34 Theorem 4» I f H i s any unimodular p o s i t i v e d e f i n i t e symmetric group m a t r i x of r a t i o n a l i n t e g e r s f o r t h e group T then H = H^H^ where i s a group m a t r i x of r a t i o n a l i n t e g e r s f o r I t i s known [3] t h a t i f H i s p o s i t i v e d e f i n i t e then 1^1 > ^* S i n c e H i s a group m a t r i x EL^ = H ^ , i = 1,*««,6« I t was e s t a b l i s h e d i n s e c t i o n 6 t h a t the group m a t r i x f o r a u n i t of f i n i t e o r d e r has a zer o d i a g o n a l . Hence t h e f o l l o w i n g r e s u l t i s c l e a r . Theorem 5« The p o s i t i v e d e f i n i t e u n i t s i n Z(S^) are a l l of i n f i n i t e o r d e r . There are i n f i n i t e l y many p o s i t i v e d e f i n i t e u n i t s of i n f i n i t e o r d e r . E x p l i c i t f o r m u l a s f o r an i n f i n i t e number of p o s i t i v e d e f i n i t e u n i t s may be found on page 1 8 . 35 B i b l i o g r a p h y 1» Go Higman, The u n i t s of group r i n g s , Proc. London Math. Soc. v o l . 46 (1940) pp. 231-248. 2> Mo Kneser, K l a s s e n z a h l e n d e f i n i t e r q u a d r a t i s c h e r Forraen, A r c h . Math. v o l . 8 (1957) PP« 76-80. 3, M. Marcus, B a s i c Theorems i n M a t r i x Theory, N a t i o n a l Bureau of Standards, A p p l i e d Mathematics S e r i e s 57. ( i 9 6 0 ) p. 3. 4« Mo Newman and 0. Taussky, On a g e n e r a l i z a t i o n of the normal b a s i s i n a b e l i a n a l g e b r a i c number f i e l d s , Comm. Pure A p p l . Math. v o l . 9 (1956) pp. 85-91. 5. H. P o l l a r d , The Theory of A l g e b r a i c Numbers, Carus Mono-graph No. 9. Math. Asso c . of America, 1950. p. 31, 6. 0. Taussky, M a t r i c e s of r a t i o n a l i n t e g e r s , B u i . Amer. Math. Soc. v o l . 66 (I960) pp. 327-345. 7. H. Co Thompson, Normal m a t r i c e s and the normal b a s i s i n a b e l i a n number f i e l d s . P a c i f i c J o u r n a l of Math. v o l . 12, No. 3 (1962) pp.. 1115-1124. 

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