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UBC Theses and Dissertations

On subgroups of prime power index Harris, L. F. 1969

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ON SUBGROUPS OF PRIME POWER INDEX by L. F. HARRIS B.A., U n i v e r s i t y of B r i t i s h Columbia, 1 9 6 6 . A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics WE ACCEPT THIS THESIS AS CONFORMING " TC/-THE REQUIRED STANDARD The U n i v e r s i t y of B r i t i s h Columbia A p r i l , 1969 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 C olumbia, 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 agree 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 o f t h i s t h e s i s f o r s c h o l a r l y p u 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 o r p u b l i c a t i o n of t h i s thes,is 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 of M a t . h M f l t i f a The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada 1 6 t h A p r i l , 1969 S u p e r v i s o r : G. K. White. ABSTRACT L e t G he t h e d i r e c t sum o f n >_ 2 c o p i e s o f t h e c y c l i c group, Z , o f i n t e g e r s . L e t p be a f i x e d p r i m e and a >_ 1 a f i x e d i n t e g e r . C o n s i d e r the subgroups, X , o f G o f i n d e x p a i n G . L e t S be a s u b s e t o f G . We say S i s a s t e l l a r  s e t i f ax e S i m p l i e s (1.1) x,2x,...,ax € S f o r any x e G and any i n t e g e r a >_ 1 . Suppose S i s a s t e l l a r s e t , p a G n S = f6 ., and S i n t e r s e c t s a l l the subgroups X o f G o f i n d e x p a i n G . We s h a l l show t h a t then |s| > p a + P a - 1 . i i i . TABLE OF CONTENTS Page 1. I n t r o d u c t i o n 1 2 . Lemmas' 7 3 . Proof of Theorem 5 •18 k. P r o o f ' o f Theorem 2 21 5 . Proof of Theorem 1 25 Appendix: N o t a t i o n , 57 B i b l i o g r a p h y ^0 I v ACKNOWLEDGEMENT I am indebted to Dr. G. K. White of the Department of Mathematics at the U n i v e r s i t y of B r i t i s h Columbia f o r d i r e c t i n g and s u p e r v i s i n g my work. \ 1. INTRODUCTION Let G be the d i r e c t sum of n >_ 2 copies of the c y c l i c group, Z , of i n t e g e r s . Let p be a f i x e d prime and a >_ 1 a f i x e d i n t e g e r . Consider the subgroups, X , of G of index p a i n G . Let S be a subset of G . We .say S i s a s t e l l a r  set i f ax € S i m p l i e s ( l . l ) x,2x,...,ax e S f o r any x c G and any i n t e g e r a >_ 1 . ct Suppose S i s a s t e l l a r s e t , p G ,D S = $ , and S i n t e r s e c t s a l l the subgroups X of G of index p a i n G . We s h a l l show tha t then |S| > p« + p*- 1 . Let • x i , X i , u i . v i , i = l , . . . , n be i n t e g e r s ; a,f3,y be p o s i t i v e i n t e g e r s . Denote p o i n t s x e G by X = ( x ^ , . . . , x^) . (a^,...,a^) = d means d i s the greatest common d i v i s o r of the a^ . Denote by X = [X - ^ ' ' ' j X n ] , (X-^,. . . , X n,p) = 1 , the subgroup X c G , [G,X] = p , such that X = {x e G | | x . x . s 0 (p a)} . I n t h e p a r t i c u l a r case a = 1 , Rogers [ 5 , p.309] n a s shown t h a t t h e n • |S| > p + 1 and White [6] has shown t h a t f o r a >_ 2 |S| > p a + p • We here e x t e n d t h i s t o o b t a i n our main r e s u l t : Theorem 1: - Suppose t h a t f o r f i x e d p a , n >_ 2 , e v e r y congruence \ n a ( 1 . 2 ) X.x = S X x s 0 . ( p u ) , (X,p) = 1 1 1 1 ! has a s o l u t i o n x = ( x p . . . 3 x n ) i n a s t e l l a r s e t S c G , and S 0 p a G = j6 . Then |s| > P a + P a - 1 . C o r o l l a r y l r Suppose p a , n >_ 2 a r e f i x e d , G ^ Z n and a s t e l l a r s e t S i n t e r s e c t s e v e r y subgroup o f i n d e x p a i n G and S n p a G = s6 . Then I«I s a , a -1 P o r a > 1 t h e theorem i s much s t r o n g e r t h a n the c o r o l l a r y s i n c e t h e r e a r e subgroups o f i n d e x p a ' 1 n o t o f t y p e X . Por example (x | S \±±x± s 0 ( p a _ 1 ) and S v i x ± = 0 (p)} 3 . has i n d e x p a when ( u ^ , . . . ,u n,p) = ( v 1 , . . . ,vn,p) = 1 and (n l 3...,U n) i- k ( v l J . . . , v n ) (p) , (k,p) =1 . There i s an i n t e r e s t i n g r e s u l t o f C a s s e l ' s [J>, p.28l] t h a t has some s i m i l a r i t y w i t h our own. F o r any a b e l i a n group G w i t h o u t elements o f f i n i t e o r d e r l e t a s t e l l a r s e t S i n t e r s e c t e v e r y subgroup o f i n d e x l,2,...,m (m f i x e d ) . Then |s| >_ m . Note t h a t Theorem 1 c o u l d be g e n e r a l i z e d a l i t t l e . As k'x s a t i s f i e s (1.2) whenever x does ( f o r any k' prime t o p ) , Theorem 1 w i l l h o l d a l s o f o r p - s t e l l a r s e t s , t h e s e b e i n g sub.sets o f S s a t i s f y i n g ( l . l ) i f a = p^ . A p a r t i c u l a r example o f a p - s t e l l a r s e t i s a p - p r i m i t i v e  s e t T , d e f i n e d as one whose p o i n t s x a r e prime t o p , T c {x e'G j (x 1,...>* n,p) = 1} = X . S i n c e we f i r s t p r o v e Theorem 1 f o r p - p r i m i t i v e s e t s , i t i s u s e f u l t o s t a t e s e p a r a t e l y t h i s s i m p l e r v e r s i o n . Theorem 2: Suppose t h a t f o r f i x e d p a , n >_ 2 e v e r y congruence n ct X»x = S X±x± = 0 (p ) , • (\,p) =1 has a p - p r i m i t i v e s o l u t i o n x e T x = ( x x , . . . , x n ) , (x.p) = 1 . , " Then |T| > p a + p " - 1 . 4. Because our c l a s s of subgroups i s defined by a congruence r e l a t i o n i t i s n a t u r a l that congruence c l a s s e s of subgroups and po i n t s should be used i n the proof of the main theorem. In p a r t i c u l a r we de r i v e an i n t e r e s t i n g r e s u l t (Theorem 5 below) concerning p - c l a s s e s of subgroups (X = X° (p)) and p o i n t s (x = x° (p)) . This r e s u l t , which analyzes fthe p - s t r u c t u r e more deeply, w i l l be needed t o prove Theorem 2 . D e f i n i t i o n : Por any p - p r i m i t i v e x° l e t A(x°) = {x e X | x = x° (p)} , (x°,p) = 1 \ be the p - c l a s s A(x°) of p o i n t s • o f X . v 2 o Theorem 3'- Let p T >_ p , n >_ j5 , X be f i x e d ; X° - [X°,... 3X°], (X°jp) = 1". Suppose that f o r each X = X° (p) the congruence (1.3) Xox = E X.x. = 0 ( p Y ) 1 has a p - p r i m i t i v e s o l u t i o n x € T . Then e i t h e r ( i ) the con-gruence i s s o l u b l e f o r each X = X° mod p by x from the subset T(x°) = T fl A(x°) f o r some x° e T -i n which case |T| > |T(x°)| > p Y " l _ or ( i i ) |T| >_ p Y _ 1 + max ( |T(x°) | , p Y ~ 2 ) f o r any x° e T . The above "or" i s the l o g i c a l one and the l a s t sentence could be r e s t a t e d : I f p Y _ 1 > |T n A( x ) | f o r a l l x e T then |T| > p ^ 1 + max (|T(x°)|a P Y " 2 ) where |T(x°)| >_ IT 0 A ( x ) | f o r a l l x e T . The problems d e a l t w i t h i n t h i s paper are examples of a more general type of problem. To s t a t e the more general problem we heed some n o t a t i o n , which we s h a l l make use of i n l a t e r s e c t i o n s . Let A be a f a m i l y of congruence c l a s s e s f\ ^  , A = £AJ_}J_€J_ i and /\^ a congruence c l a s s of sets X We may have x e \ f o r some \ e A ^ f o r some A^.. e A • We denote t h i s by square brackets: [ A:x] = [\ e U A ± | x e \) . We d i s t i n g u i s h i t from the concept e which i s defined x k A • i f f x € U \ '. We denote t h i s by round brackets: (A:x) = { A ± € A I x e A . } . [ : ] denotes a s e t o f X ; ( : ) denotes a f a m i l y o f A L e t V be a; s e t o f p o i n t s . D e f i n e [A:V] = U [A:x] xeV (A=V) = U (A:x) . xeV I f A i i s a congruence c l a s s o f X [Ai*.V-] = [X € Aj^ I x e X f o r some x € V} (Aj_:V) ={(A i } i f x f o r some x e,. V » $ o t h e r w i s e . Note: Our square b r a c k e t [ : ] i s n o t t o be c o n f u s e d w i t h t h e i n d e x o f a subgroup H i n a group G . i P o r t h e s o l u t i o n o f t h e congruences X»x,.= S X ± x i s-0 (p ) , (X,p) = 1 we may r e p l a c e X by any X mod p a and by any k'X where (k',p) = 1 . Hence w i t h o u t l o s s o f g e n e r a l i t y we may assume where c o n v e n i e n t t h a t 0 <_ \x± < p a f o r a l l i . and f o r some k U k = 1 , ( p , u ± ) >_ p f o r k < i <_ n 2. LEMMAS In t h i s s e c t i o n we prove some lemmas needed i n the proof of Theorem 3. Let A v ( n ° ) = {x | x = u° ( P a " Y ) } . Then A v ( ^ ° ) = (X = u°+np a ~ Y | u = [ p . ^ . . . 1 < 4 < P Y} and A y ^ / V ^ 0 ) = 't when m V v° ( P a " Y ) Ky^°) ^hy_±(v°) when n° H V ° ( P A " Y ) . F i x . Then /\ (u.°) i s a d i s j o i n t union of p-classes of  l a t t i c e s : • R e c a l l x feAY(P°) i f f x € U{X e A Y ( n ° ) } ... i . e . i f x e X f o r some X € A^d-*0) . ' Thus x £ A Y ( n ° ) when (1) (u° + n p a _ Y ) o x s 0- ( p a ) , u = [u-^,. . . , ^1 . This i m p l i e s /„\ o a-v / ou (2) u ox s i u l P T ( P . ) (3) u)x + n»x s 0 ( p Y ) 8. f o r some integer. m-^  and u = [u^,. . . ,\i ] mod p Y A set T c X covers /\ y(u°), [A y(n°):T] = A y(U°) , i f ( l ) has a s o l u t i o n x e -T f o r any ji e A ••• As (p. 0,p) = 1 a s u i t a b l e coordinate transformation takes u° i n t o any X . We s i m p l i f y our work b y the convenient choice u° = [ 1 , 0 , . . . , 0 ] . Denote A° = A Y(u°) . A° = (x = [ i , u 2 p a - Y , . . . , n n p a - Y ] | 1 < u . < P Y } . Then , [i 0»x. = x^ and (2) gives (4) x^ ^ = va~yw± f o r any x £ A° and (3) becomes n ( 5 ) w1- + E n i x i s 0 ( p Y ) , ( p , x 2 , . . . , x n ) = 1 - . Any c o v e r i n g of A° by T c X , [A°:T] = A ° , c o n s i s t s of p o i n t s from X° = {X € X | X = (piM^Xg, . . . , x n ) ) T° = t n x ° . T° i s a one-to-one correspondence w i t h the set T" of n-tuples ( ^ j X g j . . . , x n ) . w i t h ( x 2 , . . . ,x ,p) = 1 such that f o r every [ l , ^ , . . . , (5) has a s o l u t i o n i n £ . Note that a has now disappeared and our covering problem reduces to one i n Y o n l y . Indeed (5) i s j u s t the congruence (1.3) o f Theorem 3 w i t h u° = [1,0,...,0] . I f T c o v e r s A ° then T c o v e r s a l l the p - c l a s s e s i n A ° . Denote the s e by A ' (u') = f\ ^yi? + u ' p a ~ Y ) , u' + [u-[,.. . , 1 < \x' < p . I f x £ A'(u') then (u° +.u'p a- Y + v p a - Y + 1 ) o x == 0 ( p a ) , v = [ v 1 , . . . , v n ] . T h i s i s ( l ) w i t h u r e s t r i c t e d t o u = u' + vp , \ u' f i x e d by A ' ( n ' ) • Thus f o r u° = [1,0,...,0] [ A ' ( u ' ) : T] =/\ /(n /) i f f a l l t h e congruences n (6) ^ + T ( u ^ + v i p ) x i = 0 ( p Y ) a r e s a t i s f i e d f o r a l l u' + vp (mod p Y ) , u' f i x e d , by ( t w j _ i x 2 J " * * , x n ^ € T , ( X2^ • • • > ^ Sti ^ = 1 • ¥e need t o d i s t i n g u i s h between a p - c l a s s o f l a t t i c e s A' (P' ) and a p - c l a s s T(x°) o f p o i n t s o f T : T(x°) = {x e T | x = x° (p)} . I f \x° = [1,0,...,0]'. t h e n (.4) i m p l i e s x± s 0 (p) so (x^.,p) = 1 f o r some k > 1 . W i t h o u t l o s s o f g e n e r a l i t y t a k e k = n , x n = l , and a s u i t a b l e c o o r d i n a t e t r a n s f o r m a t i o n w i l l l e a v e u° = [1,0,...,0] f i x e d b u t t a k e x° i n t o ( 0 , . * . , 0 , l ) . Thus f o r c o n v e n i e n c e we s h a l l work w i t h 10. T* = T ( 0 , . . . , 0 , l ) but our r e s u l t s hold f o r a l l T(x°) . Now ( 4 ) and (5) imply i f x € T* and x £A Y(u°) then x e [(•pCt~yu>1,l>x2,...-i-pxn^1,l) | x± mod p Y~ 1,w 1 mod p Y} = X* . I f x = (p a _ Yu3- L,Px 2,... , p x n _ 1 , l ) £ A ° then n-1 -(7) i i ^ + P S ( n j + v i P ) x i + ^ n + v n p s 0 ( p ^  f o r some u' + vp e A° • \ . We s h a l l need subsets of. T # and sets of A ' ( l i ' ) i the lemmas. Let T c = (x e T | x = ( p a " Y c + p a " Y + 1 x 1 , p x 2 , . . . , p x n _ 1 , l ) , x i m o d p Y _ 1 } M c = {A ' ( n ' ) c A ° | y£ - -c (p)} Q ' = {Mc | [A'(u') : T j = A ' ( u ' ) f o r some A ' ( ^ ' ) e M c 5 R' = C M C I [ A ' ( n ' ) : T J / A ' ( n ' ) f o r a l l A' ( u ' ) e M c) Q' and R ' induce a d i v i s i o n of the p- c l a s s e s A ' ( P ' ) i n t o two d i s j o i n t f a m i l i e s . Q = UCA'di') e Mc. | M c e Q ' } R = U ( A ' ( u ' ) € M c | M c e R') P = {A ' ( n ' ) c A°3 = Q u R . One can get some understanding of Q and R as f o l l o w s . 1 1 . Consider a ma t r i x w i t h e n t r i e s a. . where j = c ( i n M ) and i i s determined by '' * , | J n - l i n A'd^') • L e t -1 i f [ A'(u') : T j = A ' ( n ' ) a i.1 ~ {( " i j 10 otherwise . Por example i n the m a t r i x A ! M 2 \ • 1 0 1 0 . . 0 N 0 \ 1' 0 0 . . . 0 1 0 1 0 . . 0 0 0 1 0 . . 0 0 1 1 0 . 0 1 0 1 0 . 0 the A (i-i') represented i n the. f i r s t three columns .are i n Q ; a l l the other A'(u') are i n R . I f /\'(u'.) e R then [ A ' ( n ' ) : T j ^ A ' ( ^ ' ) - • The i m p l i c a t i o n does not n e c e s s a r i l y reverse. Lemma 1: (a) I f ? -c (p) then [ A ' ( n ' ) : T c] =V . (b) To cover a p - c l a s s t\ W ) by T # we must have l T d > P Y _ 1 , "c - X (p) 12. (c) I f the p-classes /\ (p') covered come from I 3 0 <_ I = |Q'| <_ p, d i f f e r e n t T Q , then Since the number of A ' ( l - i ' ) e M c 3 f i x e d c , i s p IQI = t p n - 2 (8) |R| = p*- 1 _ ^ p n " 2 Proof: (a) I f \ x e T c , X e A'(u') then x«X H 0 ( p a ) reduces to (6) w i t h - c + px^ so c + p x 1 + P | ( p f + v.p) + ^ + v np = 0 ( p Y ) c + s 0_ (p) . (b) Since ) j = p ( Y - l ) ( n - 2 ) w e h a v e ! T C | O | [ A ' ( u ' ) : x ] | > [ A ' C u ' ) ! |TCI > P 7 " 1 . (c) The above imply |T*| > l^-1 . Since |MC| = lCA'(u') € P | ^ = - c (p)}| = p n _ 2 we get 13-n-3 Since |R| = |P| - IQI and (P| = lCA'(u') e A ° } | = p n _ 1 |R| = p - l-p R e c a l l the n o t a t i o n (P : x) = .(A'(u'). € P | x e A'(u')} Lemma 2: (a) Por any x e T • |(P : x ) | = p n " 2 . (b) I f x e T\T^ , y € T^ . then x and y are i n p p-:dasses A ' ( M ' ) : :|(P : x) n (P : y ) | = p n " 5 _ (c) I f x e. T \ T # , " y e T* then (P : y) = M c f o r some M Q so (9) |(R •: x ) | = p n - 2 - I/1'3 , I = • Proof: (a) I f x = ( p a _ Y u 3 1 3 x 2 , . . . ,x n) e A ° then x«\ = 0 ( p a ) reduces to (6). Since (x 2,....> x n,p) = 1 , suppose without l o s s of g e n e r a l i t y x 2 £ 0 (p) 14. Then we have p choices f o r each of t-t-^j • • • > H n • (b) Now suppose x e TNT^ , y e T* . Without l o s s of g e n e r a l i t y l e t x = (0,...,0,1,0) > y = (0,...,0,1) . Then x J A ' ( n ' ) ^ L I ^ = 0 ; y £ /\ ) U n = ° • T h u s x e A'(u') and y £ A ' ( u ' ) < ^ = M-n = 0 • W e have p choices f o r each of \ ^2,"'}^n-2 ' (c) I f y e T* then y e T Q f o r a unique c . By Lemma 1(a) y £ M Q f o r a unique c . Thus (P : y) = (M c i y.) Since |(P : y ) | = |(M Q : y ) | = |MC| = p n - 2 (P : y) = M c . Now |(Q : x ) | = | U / (M : x ) | = t|(M : x ) | = l\(M : x)0M | = ^|(M C :.x) PI (M c : y)|.= <,|(P : x) n (P : y ) | = . Since | (R : x) | = | (P : x) |. - J (Q : x) j = p n ~ 2 .- - t p n * 5 we have (9) . In the above lemmas we t a l k about f a m i l i e s of /\'(\x') . Induction w i l l l a t e r be used on the A'(u') € R . In p a r t i c u l a r (8) and (9) are used i n the proof of Theorem 3 . Let c,x n,...,x n be f i x e d . One p o s s i b l e 2 n-1 c T c = { ( p a _ Y c + va~y+1x1,vx2,.. . , p x n _ 1 A ) I 1 < x x In t h i s case |TCI ^ P ^ 1 and [M c :. T c] = U{X € A y I A y e Mc} and the i n e q u a l i t y of Lemma 1 becomes an e q u a l i t y . We need one f u r t h e r r e s u l t f o r Theorem'3« Lemma 3 : Let' n >_ 3 be f i x e d , T be a set of p o i n t s *X = (u) 1 3x) = (ua^Xg,... ,x n) w i t h (x,p) = 1 and T # a non-empty subset.of T , T # = [ x e T | x s x Q ( p)} , x Q f i x e d . Suppose f o r each u = [l,p. ,... ,u n] the congruence n . W l + 2 5 0 has a s o l u t i o n x e T . ' Then e i t h e r ( i ) each congruence has a s o l u t i o n x e and |T| > | T j = p 1 6 . or ( i i ) |T| > p + max ( | T j , l ) . Proof: Let /\ 2 = [ il>V-2>'' • > ^  ' 1 - u i - p ^ * T h e n j/Vj = p . Without l o s s of g e n e r a l i t y take X Q = ( 0 , . . . , 0 , l ) . Suppose |T*| < p . I f x e T then | [A 2 : x ] | = p n " 2 because x = (0,... ,0,1) e u i f f n n = 0 . Let ,y = ( 0 , . . . , 0 , 1 , 0 ) Then [A 2 : x] H [A 2 : y] = { [ l , i a 2 , . . . , L L n _ 2 , 0 , 0 ] | l'< ^  < p} so f o r any x e T x . , y e T \ T S f . | LA 2 : x] n | A 2 : y ] | = p n ~ 5 . These imply (a) | A 2 \ [ A 2 : T J | = p * " 1 - | T j p n " 2 . (h) | [ A 2:y ] V { [ A 2 : T j n [A g : y ] } | = p n " 2 - | T j p n - 5 . I f yJ" = (y^,y^,...,y^) , j = 1 , 2 are i n T\ T* then IA 2 : y 1 ] n ( A 2 : y 2 ] | ='{ pn-3 " J i J ± i f = y 2 f o r a l l i > 1 pii~"' otherwise. This together w i t h (a) and (b) gives S I fAp : y ] \ {[AP : T j n [A p : y j } j = s 1 y€T\T* 2 2 2 -X / (A 2:T #] (|T| - | T . | ) ( p n - 2 - | T j p n " 5 ) > p 1 1" 1 - | T j p n " 2 |T| - |T»| > p and |T.X. I _> 1 so T | _> p' + max (| | , 1) . \ 3 . PROOF OF THEOREM 3 To-prove Theorem 3 we proceed by i n d u c t i o n on. y • F i r s t suppose y = 2 , u° = [1,0,...,0] . Then x = ( x x , x 2 , . . . , x n ) ^ A 2 ( l i ° ) n p  x l + p I X i x i 5 0 ( p ' •j_ - 0 ( P ) » ^ = P ^ i -^ o r s o r a e ^2. x. n + £ X i x ± s 0 (p) (p<ju^,x2,... ,x n,p) = ( x 2 , . . . ,x^,p) = 1 . Thus the theorem reduces to Lemma 3 and the case y = 2 i s s e t t l e d . Suppose now that Theorem 3 holds f o r some y >_ 2 . We w i l l show i t holds f o r y + 1 . Again we may take u° = [ 1 , 0 , . . . , and the problem reduces to f i n d i n g s o l u t i o n s to the congruence n y • u)x + | s 0 (p ) w i t h x = ((ju r,x 2,.. . ,x n) such that ( x g , . . . ,x n,p) = 1 . Our proof now continues i n a s i m i l a r v e i n to that of Lemma 3 , though i t i s a l i t t l e more complicated s i n c e the r o l e played by the X e A 2 i n Lemma 3 i s played by the /\'(\x') <=A° i n Theorem 3 . The e q u a l i t i e s ( 8 ) and ( 9 ) are used as (a) and (b) were used i n Lemma 3 . In p a r t i c u l a r we must d i s t i n g u i s h two cases: 1 9 . ( 1 ) P Y > |T*| > P Y _ 1 ( 2 ) p Y _ 1 > |T(x)| f o r a l l x e T . Case 1: p Y > | T j >_ p Y _ 1 D i v i d e t h e A ' ( p ' ) c A° i n t o f a m i l i e s Q' > R', Q and R as defined i n Se c t i o n 2 . Then I = JQ'j i s r e s t r i c t e d 0 <_ l < p because p Y > | T j and |T#| _> ^ p Y - 1 by Lemma 1 ( c ) . The lemmas are a p p l i c a b l e w i t h t h i s r e s t r i c t i o n on I . P a r t i c u l a r l y note ( 8 ) and ( 9 ) . Since A'(p. 7) e R i m p l i e s [A'(u') : T j ^ A'(u') The number of p o i n t s of T i n A'(p') i s |T| _> p ^ 1 + max ( | T j , p Y ~ 2 ) f o r each A ' ( P ' ) e R .-Thus at l e a s t p Y ~ 1 p o i n t s of T \ T # are i n A ' ( p ' ) • We have E |(R : x ) | = S |{x e T\T* | x € A ' ( u ' ) } | xeT\T* A (P- ) e R (1*1 - | T j ) ( P n " 2 - t p n " 5 ) > ( p n - l - l ^ - 2 ) ^ - 1 . by ( 8 ) and ( 9 ) . |T| - | T j > P Y . |T| > p Y + max ( i T j j p Y - 1 ' ) . 20. Case 2: p Y _ 1 > |T(x)| f o r a l l x e T . By i n d u c t i o n f o r each A ' ( u ' ) e P the number of p o i n t s of T such that [A ' (u ' ) T] = A ' ( n ' ) i s |T| > p Y _ 1 + p Y ~ 2 . No t i c e JP| = p n _ 1 . Lemma 2(a) gives | (P : x ) | = p n~. 2 f o r any x e T . S l>P : x ) | = S |£x € T | x A'(U ' ) ) I xeT A (n )eP. | T | P N - 2 > p ^ 1 ^ " 1 + P Y ' 2 ) | T| >_ p Y + p Y _ 1 . For a p a r t i c u l a r T # we might take = C ( x 1 p a " Y + 1 , 0 J . . . , 0 , l ) I 1 < ^  < p 7 - 1 } and a p a r t i c u l a r A y . ^ d - 1 0 ) A Y _ i ( ^ ° ) = U I x = [ i , o , . . . , o ] ( P a - Y + 1 ) } . Then [ A ^ u 0 ) : T j = A y _ 1 ( u 0 ) and I T J - P * " 1 . 4 . PROOF OF THEOREM 2 The case a = 1 i s Roger's case [ 5 , p.309] and f o r a = 2 or n = 2 our r e s u l t f o l l o w s from [ 6 ] , So we need con-s i d e r only a >_ 3 and n _> 3 . In f a c t we assume a >_ 2 and n >_ 3 as our proof i s a p p l i c a b l e I n a l l these cases. We apply Theorem 3 w i t h a = y . Noting Theorem 3 i s true f o r any A _ (P-) we l e t /\(u) =Aa_T_(y-) i n t h i s s e c t i o n and A G = ( A ( P ) } . Since every X e U Ad-1) ^ s congruent modulo p to one of P [ 0 , . . . , 0 , 0 , 1 ] . [O,...^,!,^] -[ 0 ^  • • • j l j ^ x - j ^ ] [1,... ,x^ , x 2 , x 1 ] , 1 _< x± <_ p f o r a l l i = 1,... ,n-rl the number of d i s t i n c t A ( u ) i s n - 1 ± = S p . . o R e c a l l (J\o : x) = (A(u) e/\ c | x eA(p-)} • We have n-2 ± | (l\Q : x) | = S p f o r any x e S o s i n c e (1,0,...,0) e [ 0 , u 2 , . . . , u ] = X e U (u) . The p r o o f b r e a k s i n t o two c a s e s : Case 1: | T j >_ p 0 " 1 I f x € T # ,• y e S \ T # t h e n l ( A G : x) n (A c : y ) | = V p 1 o s i n c e (1,0,...,0) and (0,1,0,...,0) a r e b o t h , i n X i f f x x = x 2 = 0 . We have n-2 . I(A 0 : T j | = S p 1 . o T h i s g i v e s | {A(u) I fA(u) : T J = fi\ = " s 1 p 1 - ^ p 1 = p 1 1 " 1 . o o L e t M = {A(ii) | [A(n) : T j = 56} . M . i s used i n t h i s p r o o f much as R was used i n t h e p r o o f o f Theorem J . F o r each A(u) e M t h e number o f p o i n t s of" S such t h a t IA(|i) : s] =A(u) 23. i s e i t h e r |T(x)| _> p"" 1 f o r some x e S\T* 3 x t t\{\x) or | s | > p a _ 1 + max (|T(x)|, p a _ 2 } a-1 by Theorem 3. In e i t h e r case at l e a s t p p o i n t s of S\ are needed to cover each A(p-) € M . I f y € S \ T # , x e T # then |(M : y)| = | ( / \ 0 : y ) | - | (A D : y) n (AQ : x ) | = p n-2 Now ,S l ( M : y ) | = S |{y e S\T* |,y e A ( u ) 3 y e\S\T* A(n)eM n-2 v a-1 n-1 (|S| - | T j ) p n ^ > p ^ p S| - | T J > p a Case 2: p"" 1 > |T(x)| f o r a l l x e S . Since p0-"1* > |T(x)| f o r a l l x € S Theorem 3 shows at l e a s t pa""1'+pa~2 p o i n t s of S are needed f o r .each A ( n ) i n order that IA(LX) : S] = A ( u ) . Thus -S | ( A 0 : x ) | = s | {x e S | x € A ( n ) ) l ' xeS ° A(p )eA G n-2 . n-1 .• _ | s | s P 1 > .s P 1 ( P * - 1 + P A " 2 ) o o |S| > P + P Por a p a r t i c u l a r S we might choose S = { ( x , l , 0 , . . . , 0 ) | 1 < x <_ p a } u t(l,yp,0,...,0) | l<y<p' Then a l l the congruences o f Theorem 2 a r e s a t i s f i e d and so our bound i s t h e b e s t p o s s i b l e i n t h e sense t h a t i t can be a c h i e v e d . \ 5 . PROOF OF THEOREM 1 The p r o o f o f Theorem 1 i s e s s e n t i a l l y t h e same as t h a t o f Theorem 2 because (N : p^x) = (N : x) f o r N = P,Q,R,Mc and p< y . However t h e f a c t t h a t ( -L | [N : p Px] | = p P | [ N : x] |' f o r N = P3Q,R,MC, A'(ii') and P < Y r e q u i r e s some r e f i n e m e n t o f t h e lemmas. R e c a l l t h e n o t a t i o n (N : x) = {A'(n') e N | x e A ' ( n ' ) } [N : x] = {X e 'u A'(n') I x e X] N Throughout t h i s s e c t i o n we assume S i s a s t e l l a r s e t . We need a theorem on s t e l l a r s e t s c o r r e s p o n d i n g to. Theorem 3« Theorem 4: Suppose t h a t y > 2 , n > 3 and X° a r e f i x e d ; x° = [xj,...,>°] , (X°,p) = 1 and t h a t f o r each X = X (p) t h e congruence n V ( 1 . 3 ) X»x = S X x s 0 ( p Y ) 1 x has a s o l u t i o n . x e T where T i s a s t e l l a r s e t s a t i s f y i n g 26. P Y G n T .= j6 . N o t i c e T contains subsets T(x°) defined f(x°) = (my e T | y i s p - p r i m i t i v e , y = x° (p)} f o r any p - p r i m i t i v e x° e T . Then e i t h e r ( i ) the congruence i s s o l u b l e f o r each X = X° (p) by x from a subset T(x°) of T f o r some p - p r i m i t i v e x° e T i n which case . \ |T| > |T(x°)| >..pY-ls or ( i i ) |T| >. p Y - 1 + max (T(x°), p Y ~ 2 ) f o r any p - p r i m i t i v e x° e T . ~ N o t i c e T , * ( x 0 ) a r e s t e l l a r sets corresponding to the sets . T, T(x°) = T^ r e s p e c t i v e l y . For s i m p l i c i t y we w r i t e T(x ) = T^ where without l o s s of g e n e r a l i t y x° =• ( o , . . . ,0,1) . We define T n c T^ corresponding to the T c c T^ defined before T = (x e T J x = my, y = ( c p a ~ Y + x - L p a " Y + 1 , x 2 p,... ,x n_ xp x. mod p Y _ 1 3 . "c 1 N o t i c e P _ _ T~ = U T , and T i s s t e l l a r s i n c e T„ i s . * ^ c 3 c * We take without l o s s of g e n e r a l i t y H° = [1,0,...,0] as before. To prove Theorem 4 we need three lemmas s i m i l a r to those i n S e c t i o n 2. We define Q' = [M c | L A'(H ' ) : T j = A ' ( n ' ) ^ some A ' ( n ' ) e R ' = (M c | [A'(n') : T J ^ A ' ( H ' ) f o r a l l A'(u') € Q = U{A'(p.') e M c | M c € Q'} R = U {A'(n') e M c | M c e R'} . Notice P = Q 0 R . • Lemma l ' : Suppose |T*| < p Y . J » I f u£ ^ -c (p) then [ A'(n') : T J = s6 . (t>) To cover a p - c l a s s A ' ( u ' ) by we must have | T j > p ^ 1 , c 5 -p^ (p) . (c) I f the p-classes covered come.from I d i f f e r e n t Tf 0 <_ I '= |Q' | < p , then, > I P Y _ 1 ' |Q| = I p n " 2 , and (8') |R| = p 1 1" 1 - <, P n " 2 . Proof: (a) ' I f x e T , x = p^y 9 y p r i m i t i v e , then 2 8 . x«\ = 0 ( p a ) reduces to n-1 / / 6 c + p x x + p S ('^  + v ± p ) + ^ n + v np s 0 ( p y _ p ) (see ( 6 ) ) . Notice P < y since |T*| < p Y and T^ i s s t e l l a r . Therefore c + u' = 0 (p) . (b) This i s the essence of the d i f f e r e n c e between Theorems 3 and 4, and thus between Theorems 1 and 2 . In f a c t we must introduce a set v"c , not s t e l l a r , s i m i l a r to the set V i n [6, p.227], 6 — b — V = {p y e T | i f p y e T then b _< 6 , y p - p r i m i t i v e ] . ' «w 1 N o t i c e x°X = 0 (p ) f o r some x e T c i f f ^ p^y«X s 0 ( p a ) f o r some p^y e v"c so that [A ' (n ' ) :• T c] = [A ' (n ' ) : V c] . A l s o p^y , p°uu e V"c i m p l i e s y t «> ( p ) so {my | m = l,...,p^} n (muu j m = l, . . . , p 0 } = $6 which shows | T J > I P P . P 3 y e v c . . 29. Let a = p^ Y 1 ) ( n~ 2) = |[A ' ( | i ' ) : x ] | f o r any p - p r i m i t i v e x e A'(u') • Then i f p P y e V Q , |[A'(n') :.p Py.]| = P P a so | T C | > E P P = E l [ A # ( ^ ) ' P P y 3 l , | E | [ A > ' ) : p p y ] | [A'(P.') = V J | = i | [A ' ( | i ' ) : T J | = JAlliiliL = pY-1 Thus |T | > p Y _ 1 ai \ \r- / c a 1 ^ r ' c a P _ (c) Since T* = U T^ we have \ T J = E |T j ^ I r j 7 " 1 by (b) above. . 1 c Since the number of A'd-1') i n M i s c n - 2 we have !'MC | = p IQI = I P n _ 2 . F i n a l l y |P| = pn~^" and P i s the d i s j o i n t union of Q and R so |R| = |P| - |Q| = V 1 " 1 - I p n " 2 ' Lemma 2' : I f | T J < p Y and p YG n T = 6 then (a) j ( P : x ) | = p n " 2 f o r any x e T (b) I f x e T\T* , y e T* then ' |(P : x) n (P : y ) | = p n " 5 3 0 . ( c ) I f x,y as i n (b) t h e n (P : y ) = M c i m p l i e s (9') |(R : x ) | = p n " 2 - I p n " 3 , I = |Q'| . P r o o f : N o t i c e i n t h i s lemma we t a l k about (N : x) , N = P,Q,R and r e c a l l we mentioned a t the b e g i n n i n g o f t h i s s e c t i o n t h a t t h e s t e l l a r p r o p e r t y does n o t a f f e c t t h i s r e l a t i o n . (a) I f \ x e T , x k A ° t h e n x»\ = 0 ( p a ) reduces t o w l + | K + v i p ) x i H 0 ( p Y ~ ^ ' by ( 6 ) , where x = p&($a~ym1,x2,...,xn) ( x 2 , . .. , x n , p ) = 1, £i < y s i n c e | T j < p Y . Hence w i t h o u t l o s s o f g e n e r a l i t y t a k e x g ^ 0 (p) . Then we have p c h o i c e s f o r each o f P-^J • • • y^Yi * N o t i c e t h i s means (p,y) = ( P , P P y ) • (b) I f x e T N T * , y e T* w i t h o u t l o s s o f g e n e r a l i t y t a k e x = p P ( 0 , . . . , 0 , 1 , 0 ) , p < Y ... y = p 6 ( 0 , . . . , 0 , 0 , 1 ) , 6 < Y • Then x e A' ( u ' ^  p ^ _ 1 = 0 , y e A' (u' )*>P-^ = 0 so" x £ A'(u ') and y £ A ' C v 1 ' ) ^ P - n _ i - ^ n = 0 • W e h a v e p c h o i c e s f o r ^2 ^ 3 5 " " * n-2 * >,x) n ( P , y ) | = P n " 5 . . ( c ) I f y € T* , y e T f o r u n i q u e c . By Lemma l ' ( a ) , y e M f o r u n i q u e c , so t h a t (P,y) = (M ,y) c M . l ( p a y ) l = | M C | = P N " 2 . i m p l i e s ( P 5 y ) = M c . \ • Now |(Q : x ) | = | S (M : x ) | = l\(U : x ) | M ceQ' C c = * | ( M C : x) n MCT = l\(MQ : x) H (M Q : x)| = l \ ( P : : x) n (P : y ) | = I p n ~ 5 |(R : x)| = |(P : x ) L - |(Q : x)| = P n " 2 --1 v n - 3 • Lemma 3': Suppose n >_ 3 and f o r each \ = [ l , X 2 , . . . ,X n] the congruence ' n UJ-, + £ X.x. = 0 (p) 2 ' has a s o l u t i o n x e T , where T i s a s t e l l a r s e t o f p o i n t s o f ty p e x = my = m(u)^,y) , (y,p) = 1 32. N o t i c e T c o n t a i n s s u b s e t s T(x°) d e f i n e d f(x°) = {my e T | y = ( u ^ y ) , y = x°(p)} f o r x° e T . Then e i t h e r ( i ) each congruence has a s o l u t i o n x from a T(x ) i n w h i c h case |T| > |T(x°)| > p . o r ( i i ) |T| >_ p + max (|T(x°)|,l) f o r any x° e T . P r o o f : I f -x |T(x°)| >_ p f o r some x° € T we a r e done. Assume |T(x°)| < p f o r a l l x° € T . I f p P y € T , t h e n i P P y , ( p P - l ) y , . . . , 2 y , y e T ( y ) so jT(x°)j < p f o r a l l x° € T i m p l i e s T i s a p - p r i m i t i v e s e t . Lemma 3' thus reduces t o Lemma 3, w h i c h we have- a l r e a d y proven.. -Now we a r e r e a d y t o p r o v e Theorem 4. W i t h o u t l o s s o f g e n e r a l i t y we t a k e as b e f o r e u° = [1,0,...,0] x° = (0,...,0,1) . Suppose Y = 2 . Then x e A 2(^°) i f f n .„ x 1 + p | \ ± x ± s 0 (p ) x± = 0 ..(p) x l = p t u i n wl + | X i x i U 0 ^ 3 ( p w 1 3 x 2 , . . . ^ x ^ p ) *= ( x 2 , . . . j X ^ p ) = 1 . Thus the theorem reduces t o Lemma 3' . We assume t h e theorem t r u e f o r y and p r o v e i t f o r Y + 1 . Case 1: p Y > | T j >_ p Y _ 1 By i n d u c t i o n each A ' ( M . ' ) i n R w i l l need a t l e a s t p Y _ 1 p o i n t s o f T \ T * such t h a t [A'(n') : T] =A'(!-0 • Note ( 8 ' ) and ( 9 ' ) . We have E |(R :• x ) | = E |(x e T \ V | x £/\'(u')}| xeTNT* f\(\\')eR (|T| - | T j ) ( P n " 2 - t p n " 5 ) > ( p ^ 1 - I p n - 2 ) p Y - 1 |T| - | T j > p Y |T| > p Y + max ( | T j , p Y _ 1 ) Case 2: | T ( x ) | < p Y - 1 f o r a l l x e T . By i n d u c t i o n each /\'(u') e P w i l l need a t l e a s t p Y _ 1 + p Y 2 p o i n t s of T such that [A'(n') : T] = A'(ui') • R e c a l l Lemma 2 ' ( a ) : |(P : x ) | = p n " 2 f o r any x e T. , and note We have E H(P : x ) | .= Z |{x € T | x £ A ' ( u ' ) l xeT A'Cp/JeP n-21 mi \ / Y - l , J - 2 \ n-1 |T| > ( p ^ + P Y - < V T| > p Y + p ^ 1 N o t i c e how the proof of Theorem 4 i s s i m i l a r to the proof of Theorem 3« S t e l l a r subsets of S are used i n place of p - p r i m i t i v e subsets of S ; ( 8 ' ) , ( 9 ' ) are used i n the place of ( 8 ) and (9) and the Lemmas ' are used i n pl a c e of the Lemmas. Theorem 1 r e l a t e s to Theorem 2 i n the same manner. We now prove Theorem 1. We apply Theorem 4 w i t h Y = a > 2 , n > _ 3 . R e c a l l the n o t a t i o n A(|i) = {X I X = ii(p)} A 0 = CA(^)) . In the l a s t s e c t i o n we saw 35. n-1 , I A J = * P 1 • o I f x e S , without l o s s of g e n e r a l i t y take x = p p ( l , 0 , . . . , 0 ) J ( [ 0 , u 2 , . . . 3 p . n l ) so n-2 : x ) | =' S p 1 f o r any x e S . o . As before the proof breaks i n t o the two cases ( l ) |T*| > P a _ 1 and (2) p a - 1 > |T(x)| f o r a l l x e S . Case l : | T j >_ p " - 1 . Let M = (A.(u) e A 0 I [A(u) : T j = f6) and r e c a l l n-1 . n-2 . -, |M| = I A J " l ( A 0 : = T, p x - £ V = P " • By i n d u c t i o n ° ° a 1 each A(u) e M w i l l need at l e a s t p p o i n t s of S\T* i n order that • [A(p.) : S] =A(n) . I f y = p 5 ( 0 , . . . ,0,1) e ( [ ^ , . . . , ^ ^ , 0 ] ) and x, as above then (AQ : x ) n (J[Q : y) • = {A( [ 0 , ( i 2 , . . . , u n _ 1 , 0 ] ) } so n-3 , _ | (A : x) n (A- : y ) | = S P f o r any x e S \ T # , -o y e and (AQ y ) = (A 0 : ?*) = A 0 \ M • l (A G : x) n (A 0\M}| + | (A 0 : x) n M | = | A o | |M 0 (Ao : x ) | = |(M : x ) | = p n " 2 , 36. We have S - |(M : x ) | = £ |{x e S \ T* | x e A ( u ) 3 | xeSNT* A(P-)eM t i s i - iT.i)p»- a > p ^ y * - 1 |S| - | T J > p° Case 2: p a 1 > | T ( x j | f o r a l l x e S . By i n d u c t i o n each A(p-) e A Q w i l l r e q u i r e at l e a s t •pa~^ + p a ~ 2 p o i n t s of S i n order that [A(u) : S] = A(u) . We have S l ( A 0 : x ) | = s |{x £ S | x e A ( n ) } | xes ° A(u )eA 0 Isl ni2 P 1 > I A O K P " - 1 + P°-*> = " V ^ " ^ " 2 ) 0 o 1 i a a-1 |S| > p U + p Notice the s t r i c t i n e q u a l i t y i n Case 2; the same one appeared i n the proof of Theorem 2. APPENDIX: NOTATION J G = Z , n _> 2 p prime J a >_ 1 S s t e l l a r s e t x = (x. ,.. . , x ) € G n . . x = [ x 1 , . . . , x n ] = (x € G I S \±x± s 0 ( p a ) } , (X,p) = 1 X = {x e G | (x,p) = 1} n X«x = Y. X.x. 1 1 1 : • . A(x°) = {x € X | X = x° (p)} x e A Y ( u ° ) ~ * c U{X £ A Y ( ^ ° ) } u° ,""u , v°,v , u' a r e subgroups o f ty p e X . IA Y(u°) : T] = {X e A Y ( ^ ° ) I x e X f o r some x e T} A ° = A Y ( ^ ° ) = U = [ i ^ / - ^ . . ^ / ' " ] I 1 < M ± < P Y ) u° = [1,0,...,0] A ' ( P ' ) = / V J > ° + ^ ' p a " Y ) i s a p - c l a s s o f A ° T , a s e t w h i c h c o v e r s /\° T(x°) = {x € T | x = x° ( p ) ) x° = (0,...,0,1) -T(x°) = T* = T(0,...,0,1) T c = {x £ T | x = ( p a " Y c + p ^ ^ ^ v p x g ^ . ^ p x ^ ' , ! ) | x ± mod p Y _ 1 } M C = CA'(n') U A - -c (p)} 3 8 . Q' = [ M C I [ A ' ( u ' ) : T j = A ' ( n ' ) f o r some. A ' ( n ' ) e R ' = ( M C | [ A ' ( n ' ) : T J ^ A ' ( u ' ) f o r a l l A ' ( P ' ) * M C 3 Q = UCA' (^ # ) e M c | M c"e Q'} R = U l A ' ( n ' ) e M c I M c e R'} P = Q U R [A'(vi ' ) : T c ] = (\ e/\'(^') I x e X f o r some x e T Q} (P : x) = (A ' ( u ' ) e P I x € A ' (w ' ) } x = ( x ^ , . . • , x^) A 2 \ t A 2 : T * ] X = [\ € A 2 I X i [A 2 : T J ) A 2 = { [ l , ( i 2 , . . . , P n ] I 1 < ^ < p) i A a _ ; i > ) = {X | X H u (p)} A 0 = C/I(P03. ( A Q : x) = CA(ut) e A Q I x k A ( u ) } (A(P-) : S ] = U e A ( u ) I x E X f o r some x € S ] M = (A(u) I [A(M) : T J = rf) (M : y) = (A(n) e M | y k A ( u ) 3 T , s t e l l a r s e t c o v e r i n g A ° = (X = X° (p ) 3 T(x°) = (my e T | y i s p - p r i m i t i v e , y = x° (p)} T* = T ( 0 , . . . , 0 , 1 ) T c = (my € j y = (cj> a~ y+x 1p a~ y + 1,x 2p,... , ^ ^ , 1 ) ,x± mod p 7 " 1 } Q' = ( M C | [A ' (p ' ) .: T J = A ' ( n ' ) f o r some A ' ( u ' ) e M Q 3 R' = (M C I [A'(u') : T J / A ' (n ' ) for a l l /\'(u'') e M C} Q = U{A'(M') e M C | M C e Q'} R = UlA ' (u ' ) € M C | M C £ R'} A ^ n 0 ) = ( \ U ^ ° ( P a " Y ) } \ BIBLIOGRAPHY 1. MacLane, S., and B i r k h o f f , G., Algebra, Macmillan, New York, 1967. 2. Cassels, J.W.S., An I n t r o d u c t i o n to the Geometry of Numbers, Spri n g e r - V e r l a g , B e r l i n , 1959. 3. C a s s e l s , J.W.S., "On the subgroups of I n f i n i t e Groups", J.L.M.S., 33 (1958), pp. 281-284. 4. Hardy, &.M., and Wright, E.M., An I n t r o d u c t i o n to the Theory of Numbers, 4 t h Ed., Clarendon Press, Oxford, i 9 6 0 . 5. Rogers, C.A., "The Number of L a t t i c e P o i n t s i n a Star Body", J.L.M.S., 26, (1951), pp. 307-310. 6 . /" White, G.K., "On subgroups of f i x e d index", Pac. J . Math., 28, (1969), PP. 225-232. 

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