A SURVEY OF RECENT RESULTS on. • i TORSION FREE ABELIAN GROUPS by STANLEY HOWARD DUKE B.Sc., McMASTER UNIVERSITY, 1964 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF • MASTER OF ARTS i n t h e Department o f MATHEMATICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1967 In presenting this thesis in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the Library shal l make i t f ree ly avai lable for reference and study, I further agree that permission for extensive copying of th is thesis for scholarly purposes may be granted by the Head of my Department or by his representatives, It is understood that copying or publ ica t ion of th is thesis for f i n a n c i a l gain shal l not be allowed without my wri t ten permission. Department of MflTHEMfrTjC^ The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date iffULUf M l ABSTRACT T h i s t h e s i s i s a survey o f some r e c e n t r e s u l t s con c e r n i n g t o r s i o n f r e e a b e l i a n g r o u p s , h e r e a f t e r r e f e r r e d t o a s groups. The emphasis i s on c o u n t a b l e groups,, p a r t i c u l a r l y groups of f i n i t e r ank. S e c t i o n 1 c o n t a i n s t h e i n t r o d u c t i o n and some n o t a t i o n used t h r o u g h o u t t h i s t h e s i s . We b e g i n i n s e c t i o n 2 by d e s c r i b i n g the g e n e r a l n a t u r e o f t h e e x i s t i n g c h a r a c t e r i z a t i o n s f o r c o u n t a b l e groups and by d e s c r i b i n g why t h e s e c h a r a c t e r i z a t i o n s do n o t p r o - v i d e s a t i s f a c t o r y systems o f i n v a r i a n t s . We i n c l u d e here a b r i e f d e s c r i p t i o n o f a c l a s s i f i c a t i o n f o r groups o f a r b i t r a r y power.' P a t h o l o g i e s o f groups a r e d i s c u s s e d i n s e c t i o n 5. We b r i e f l y d i s cuss rank.one groups and c o m p l e t e l y decomposable groups and t h e n p r e s e n t examples t o show the v a s t number o f i n d e c o m p o s a b l e groups w h i c h e x i s t and t h a t a group may have two d i f f e r e n t d e c o m p o s i t i o n s i n t o the d i r e c t sum o f indecomposable groups.' Quasi-Isomorphism"~" and the r i n g o f quasi-endomorphisms of a group a r e i n t r o d u c e d i n ' \ s e c t i o n k- and d i s c u s s e d b r i e f l y . We p r e s e n t t h e theorems w h i c h \ e s t a b l i s h t h e i m p o r t a n c e o f t h e s e n o t i o n s ; namely t h a t ( i ) q u a s i - 1 d e c o m p o s i t i o n s o f c e r t a i n groups a r e u n i q u e up to' q u a s i - I s o m o r p h i s m and ( i i ) t h e q u a s i - d e c o m p o s i t i o n t h e o r y o f c e r t a i n g r o u p s i s e q u i v a l e n t t o t h e d e c o m p o s i t i o n , t h e o r y o f t h e quasi-endomorphism r i n g c o n s i d e r e d as a r i g h t module o v e r I t s e l f . I n c l u d e d under ' c e r t a i n groups" a r e t h e groups o f f i n i t e r a n k . S e c t i o n 5 i s devoted t o rank two gro u p s . We o u t l i n e t h e development o f the q u a s i - i s o m o r p h i s m i n v a r i a n t s f o r r a n k two groups, due t o Beaumont.and P i e r c e , and d i s c u s s ' some o f t h e i r ( i i ) ( I l l ) a p p l i c a t i o n s . F o r example, c o n d i t i o n s , i n terms o f t h e i n v a r i a n t s , are g i v e n f o r q u a s i - i s o m o r p h i c rank two groups t o be i s o m o r p h i c . ^ Type s e t s a r e reviewed i n s e c t i o n 6. We p r e s e n t b o t h n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s - f o r s e t s o f t y p e s t o be t h e t y p e s e t s o f rank two groups and o f groups o f a r b i t r a r y f i n i t e r a n k . We devote s e c t i o n 7 t o a b r i e f d i s c u s s i o n o f t h e n o t i o n and i m p o r t a n c e of q u a s i - e s s e n t i a l groups. The ideas-'of i r r e d u c i b i l i t y and t h e p s u e d o - s o c l e a r e d e f i n e d i n s e c t i o n 8. > We d e m o n s t r a t e how t h e s e i d e a s a f f e c t the s t r u c t u r e o f t h e quasi-endomorphism r i n g by showing how t h e y can be used t o c o m p e t e th e quasi-endomorphism • r i n g o f rank two groups. ( i v ) ACIO^OITLEDGSMENTS I w i s h t o e x p r e s s my thanks, t o D r . Roy W e s t w i c k f o r t h e t i m e , p a t i e n c e 3 ...and i n v a l u a b l e a s s i s t a n c e he gave t o t h e w r i t i n g o f t h i s t h e s i s . I a l s o w i s h t o e x p r e s s t h a n k s t o t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a and t o t h e Canada C o u n c i l f o r t h e i r k i n d "N f i n a n c i a l a s s i s a t n c e t o me w h i l e w o r k i n g t o w a r d s my M.A. / / /' / TABLE OF CONTENTS S e c t i o n 1 I n t r o d u c t i o n . . 1 S e c t i o n 2 Isomorphism i n v a r i a n t s f o r c o u n t a b l e groups 3 S e c t i o n 5 I n d e c o m p o s a b i l i t y a n d \ d i r e c t summation 16 S e c t i o n 4 Qua s i - i s o m o r p h i s m and quasi -endomorphisms 26 S e c t i o n 5 Rank tv;o groups • 3^ S e c t i o n 6 Type s e t s 43 S e c t i o n 7 Q u a s i - e s s e n t i a l groups 57 S e c t i o n 8 I r r e d u c i b i l i t y , the p s e u d o - s o c l e and t h e 60 r i n g of quas i -endomofphisms ' S e c t i o n 9 B i b l i o g r a p h y ,64 INTRODUCTION The problem of c l a s s i f y i n g countable a b e l i a n groups by a complete set of i n v a r i a n t s has not y e t been s o l v e d . S a t i s f a c t o r y complete systems of i n v a r i a n t s have been found f o r some s p e c i a l c l a s s e s of a b e l i a n groups such as the t o r s i o n groups and the rank one t o r s i o n f r e e groups. However the mixed c o u n t a b l e a b e l i a n groups have not y e t been c l a s s i f i e d . Every mixed group can be regarded as an e x t e n s i o n of a t o r s i o n group, namely i t s t o r s i o n p a r t , by a t o r s i o n f r e e group. Hence the c l a s s i f i c a t i o n of t o r s i o n f r e e groups p l a y s an important p a r t i n the c l a s s i f i c a t i o n of mixed groups. C e r t a i n systems of i n v a r i a n t s do e x i s t f o r the countable t o r s i o n f r e e a b e l i a n groups. These systems p r o v i d e schemes f o r the c o n s t r u c t i o n of such groups and the groups o b t a i n e d are isomorphic i f f the schemes a r e e q u i v a l e n t i n some sense. The^ schemes are u s u a l l y i n terms of m a t r i c e s and the e q u i v a l e n c e -\ . . . ' • : \ problem f o r the schemes i s unsolved. As a consequence, "these i n - \ v a r i a n t s do not p r o v i d e a s a t i s f a c t o r y c h a r a c t e r i z a t i o n . The purpose of t h i s t h e s i s i s t o survey some of the work t h a t has been done i n the study and c l a s s i f i c a t i o n of t o r s i o n f r e e a b e l i a n groups and to present resumees of the main r e s u l t s . As a r e s u l t , very few of the theorems w i l l be proved and none w i l l be proved I n d e t a i l . Complete d e t a i l s of p r o o f can, of course, be found i n the papers i n d i c a t e d . The papers reviewed here will';, f o r the l a r g e p a r t , be papers presented s i n c e the p u b l i c a t i o n i n 1958 of L. Fuchs book " A b e l i a n Groups". The emphasis w i l l be on t o r s i o n - f r e e a b e l i a n groups of f i n i t e rank. 2. Throughout, t h i s t h e s i s we w i l l use t h e f o l l o w i n g n o t a t i o n : Z: t h e domain o f r a t i o n a l i n t e g e r s H : (n e Z.| n > 0] u { « } R : the f i e l d o f r a t i o n a l numbers Z,( p): The domain of p - a d i c i n t e g e r s R ( P ) : t h e f i e l d o f p - a d i c numbers R = ( r e R | (b^p) = 1} where p i s a r a t i o n a l . p r i m e R N : an n - d i m e n s i o n a l r a t i o n a l v e c t o r space Tf : t h e s e t o f r a t i o n a l p r i m e s v.. [ 8 ] : t h e t y p e c o n t a i n i n g t h e c h a r a c t e r i s t i c ; I f G > G a a r e groups and x e G t h e n we w i l l use TT G„ : t h e c a r t e s i a n p r o d u c t o f t h e G where a v a r i e s o v e r some "a a K a i n d e x s e t . h ( x , G ) : t h e p - h e i g h t o f x i n G h(x,G) : t h e h e i g h t o f x i n G ; i . e . h(x,G) (p_) - h (x,G) t ( x j G ) : t h e t y p e o f x i n G . Where t h e r e i s no chance o f c o n f u s i o n we w i l l w r i t e t ( x ) . : T(G) : t h e t y p e s e t o f G' i . e . T(G) - ( t ( x , G ) | x e G} G t = [2 £ G | t(g,G) > t } where t i s a t y p e r(G) : t h e rank o f G . p. . I f G i s a t o r s i o n f r e e a b e l i a n group we w i l l u s e V t o denote t h e m i n i m a l d i v i s i b l e group c o n t a i n i n g G . V can be c o n s i d e r e d as a r a t i o n a l v e c t o r space w i t h d i m e n s i o n e q u a l t o r(G) . Hence G I s a f u l l subgroup o f V ( a group A i s a f u l l subgroup o f a t o r s i o n f r e e a b e l i a n group B i f B/A I s a t o r s i o n g r o u p ) . A l l t o r s i o n f r e e a b e l i a n groups w i t h r a n k < r(G) can be considered' as subgroups o f V . I f r ( G ) = n , we w i l l sometimes f i n d i t c o n v e n i e n t t o w r i t e V =» R n . Through out t l i i s t h e s i s we w i l l use t h e word gro u p , u n l e s s o t h e r w i s e s p e c i f i e d , t o denote a t o r s i o n / f r e e a b e l i a n group and t h u s a f u l l sub- / ^roup o f a r a t i o n a l v e c t o r space w i t h d i m e n s i o n e q u a l t o the r a n k of the groups By a b a s i s / o f a group, we w i l l mean a maximal l i n e a r l y •• • / ' Independent set o f elements o f t h e grou p . A b a s i s o f a group G w i l l a l s o be a b a s i s , i n t h e u s u a l s e n s e , o f t h e v e c t o r / space V . 2. Isomorphism I n v a r i a n t s f o r C o u n t a b l e Groups. As we mentioned I n t h e I n t r o d u c t i o n , c e r t a i n systems of i n v a r i a n t s d o / e x i s t f o r t h e c o u n t a b l e g r o u p s . These systems u s u a l l y a r i s e - a,s f o l l o w s . A. b a s i s o f a c o u n t a b l e group G i s s e l e c t e d and used t o d e r i v e a scheme o f i n v a r i a n t s f o r t h e g r o u p . Then i t i s shown t h a t s uch schemes can be used t o c o n s t r u c t a l l c o u n t a b l e groups, i n some c a s e s o f f i n i t e r a n k , and i n o t h e r c a s e s of b o t h f i n i t e and I n f i n i t e rank. However, d i f f e r e n t b a s e s o f a group g i v e r i s e t o d i f f e r e n t schemes 'and d i f f e r e n t schemes can be used t o c o n s t r u c t i s o m o r p h i c g r o u p s . To r e c t i f y t h i s > a n e q u i v a - l e n c e I s then d e f i n e d on t h e schemes so t h a t e q u i v a l e n t schemes w i l l c o r r e s p o n d t o i s o m o r p h i c g r o u p s . The e q u i v a l e n c e c l a s s e s / of the schemes then form a complete system of i n v a r i a n t s f o r the gz-oups. Since the schemes are usually i n terms i n matrices, the systems of invariants consist of c e r t a i n equivalence classes of matrices. I t i s here that the problem with these systems a r i s e s . As Fuchs L13J "has shown, i t Is possible to determine which equivalence classes of matrices correspond to countable groups. However the problem of a c t u a l l y determining the equivalence classes has not been solved. Hence these various systems do not give satisfactory characterizations of the countable groups. These systems are valuable, though, i n that they provide methods of describing the countable groups and thus deepen our knowledge of t h e i r structure. They have also provided methods f o r construct ing nev; examples of indecomposable groups. The f i r s t of these systems of i n v a r i a n t s were provided by Kurosh [24] , Derry [ 9 ] , and Mal'cev [ 2 7 ] . Kurosh found invariants for primitive groups of f i n i t e rank and Derry and Mal'cev presented invariants f o r a r b i t r a r y groups of f i n i t e rank. A l l three used c e r t a i n equivalence classes of I n f i n i t e sequences of f i n i t e matrices of p-adic numbers to describe the groups. Kurosh's c l a s s i f i c a t i o n provided the f i r s t examples of indecomposable groups Jof a r b i t r a r y f i n i t e rank. The only previous examples of Indecomposable groups were rank two groups found by Levi [26] and Pontryagin [ 2 8 ] . The r e s u l t s of Kurosh, Derry and Mal'cev have been generalized In Fuchs [lj5] to provide i n - variants^ i n terms of i n f i n i t e matrices, f o r a l l countable groups of both f i n i t e and i n f i n i t e rank. Szekeres [ 3 6 ] has'given another c l a s s i f i c a t i o n f o r a r b i t r a r y countable groups i n terms of certain p-adic and integral invariants. However this c l a s s i f i c a tion i s not complete for Szeker'es does not study the effect of a change in basis or determine conditions for equivalence of his schemes. Two other systems have been developed by Campbell [ 5 ] and Rotman [ 3 5 ] , who use somewhat similar approaches. Campbell' invariants are for arbitrary countable groups whereas Rotman's'.- are for f i n i t e rank groups only. Campbells approach i s based on the fact that a group i s determined once, the d i v i s i b i l i t y pro perties of a basal subgroup (a subgroup generated by a basis of a group)are known. The schemes produced are certain systems of sequences of additive groups of suitable ordered sets of integers and are called D-systems . The equivalence classes of D-iystems under a suitable equivalence relation are called D-types. Rotman, instead of considering basal subgroups, considers, ordered bases of a group with f i n i t e rank. A generalized height function i s defined and an equivalence relation i s set up on these functions. To i l l u s t r a t e we w i l l now examine the invariants of Szekeres, Campbell," and Rotman. Szekeres was the f i r s t to give invariants for arbitrary countable groups. Campbell's and Rotman's invariants, are the most recent ones presented. We w i l l also examine a system of invariants developed by Erdos [ 1 2 ] . T l i i s system i s not for the countable groups only but for groups of arbitrary power. Erdos uses torsion free factor groups of free abelian groups to classify groups of arbitrary power with in f i n i t e matrices. Hence i t i s appropriate to consider Erdos' • system here. Szekeres makes use of the notions of 'independence and dependence modulo p and p where p e T . Let g n j . . . j g i , he an independent set jot elements of a group G . Then we say- that g-,,...gv are dependent modulo p n (n i s a p o s i t i v e integer) 1 k y • - v i f y k a.g.' = 6 (mod p n) has a solution with at l e a s t one (a^p) = 1 where a l l are Integers. Otherwise g-^.-.g^ w i l l he called independent modulo p n . Now suppose that a e has the standard representa tion a = a Q +• a^p + a 2 p 2 + . .. , where 0 _< a i < p f o r a l l i . Then we w i l l write a^n^. » a„ + a.p + ... + a„ , p n _ 1 . I f o 1 n-1 a, a *'e z ( p ) , we w i l l write , E^ -, a, g. = 0 (mod p°°) to i n d i c a t e X K ' , . . - i X X X=X that £lc .oJ n^- g, s 0 (mod p n) f o r every n . I f there e x i s t ... i = l 1 1 p-adic integers a-^3... , not a l l zero, such that Z a.g. =0 (mod p ) . then v/e w i l l say that ,g n,...,g v are i = l 1 7 ' dependent modulo p°° i~. Otherwise we w i l l say that they are inde pendent modulo p" . The construction of the Invariants employs the following .lemma'. c • Lemma I f g-j^.^g^' are independent modulo p and g i s independ ent of g-j_j...jgj c modulo p 0 0 , then g. cannot he dependent on SjL > • • • * Sfc modulo % p n f o r large enough n . We now construct the i n v a r i a n t s . Let G he a countable group .and S = {a^,a2,...) he a basis of G . Choose i some p e ir . Let a." be the f i r s t element of S that i s not inde- r l l pendent modulo p 0 0 ( i x need not exist) . I f i ^ e x i s t s then, by the lemma, there i s a largest h^ with a i = 0 (mod p ) . We write p xb- » a , b n « p a. and construct i n d u c t i v e l y 1 • 1 a sequence b 1,b 2,... of elements that are Independent modulo p • b, has already-been constructed. Let a. be the f i r s t element l x R of S that i s l i n e a r l y independent of b^,..., b^_ 1 modulo p 0 0 . Then there i s a greatest h^ and uniquely defined c o e f f i c i e n t s H. . k-1 . . h, 0 < a. < P ({=1,...,k-l) with a, s E b»(mod p ) . We ~ xk,i x k <U1 v-.-ti :k-1 ., write b = p (a, — £ a, b^) . The b, s are constructed u n t i l a l l the a. s are iv X r exhausted. The number, s , of s . i s f i n i t e or i n f i n i t e and does not depend-on the choice of S . We c a l l s = s(p) the rank of G modulo p" . For a l l p we have 0 < s(p) X r = r(G) Let M(p)„ denote the set of i n d i c e s 1^ , » i k ( p ) (1 .< k _< s(p)) , and Njp) denote the set of i n d i c e s j $ M(p) (j5- < J < r) • Then N(p) => fi i f s(.p) = r and i k m k f o r every k _< r . Suppose N(p) ^ <P and j" e N(p) . Writing i Q = 0 i g + 1 =• r + 1, we then have < j < i k + x f o r s o r n e ^ ° S• k S s • 6 a . . s ^ p i h i (mod p w) where (3,p 1,... ,3^ ... e Z^Pi . Not a l l of\ Nov; a. Is independent of b 1...b, modulo p™ and hence \ V . J X K . • \ ^ (p) ... Y 8. the 8,B^,...,B k a r e = 0 (mod p) and i n p a r t i c u l a r p t'O (mod p) . Hence we have a. = s k a.vb, (mod p0 9) f o r J i = l «*' 1 some a..,....,a., e and u n i q u e l y d e t e r m i n e d by a . . Because of t h i s l a s t c o n g r u e n c e t h e r e e x i s t s an i n f i n i t e sequence, b j r t ( n = 0 , l , 2 , . . . ) , o f elements o f G w i t h b ^ o » &y b j V - = p - n ( a . - z k a\*\) . J i = l " " Thus we see t h a t e a c h b a s i s o f G u n i q u e l y d e t e r m i n e s a system o f i n v a r i a n t s M(p) » [ I k ( p ) l , h^p) (k...- l , . . . , s ( p ) ) (I) " 'a J 4(p)- < (J >'-i t(p)) f o r e v e r y p e TT . There systems a r e s u b j e c t t o t h e c o n d i t i o n s 0 < s(p) < r , h k ( p ) > 0 c' (ID 0 < « l k | t ( p ) < P M W ( i k 6 M(p)) Szekeres t h e n d e m o n s t r a t e s t h a t h i s systems o f i n v a r i a n t s : ; can be used t o d e s c r i b e a l l c o u n t a b l e g r o u p s by p r o v i n g ; Theorem 2.1 The s e t o f elements [ b k ( p ) , b ^ n ( p ) ] g e n e r a t e s t h e group G and e v e r y element o f G can be e x p r e s s e d u n i q u e l y , i n t h e form . . . s(p) , x a. + £ £ y k ( p ) \ ( p ) + E E £* Z i n ( p ) b i n ( p ) where 0 < y k ( p ) < p ^ ^ / 0 < Z J n ( p ) < p , ^ e Z . c Theorem 2 .2 I f , f o r e v e r y p e ir , an a r b i t r a r y system-' ( I ) i s g i v e n , s a t i s f y i n g c o n d i t i o n s ( I I ) , t h e n t h e r e i s e x a c t l y one group G b e l o n g i n g t o t h i s sytem o f i n v a r i a n t s ( i . e . no two non - i m o r p h i c groups b e l o n g t o t h i s s y s t e m ) . , These two theorems a l l o w c o n s t r u c t i o n and c h a r a c t e r i z a t i o n o f a l l c o u n t a b l e g r o u p s . However, a s we mentioned above, s i n c e a b a s i s change i s n o t c o n s i d e r e d and no c o n d i t i o n s f o r t h e e q u i v a l e n c e ' o f two systems a r e f o u n d , t h e c h a r a c t e r i z a t i o n i s n o t complete/' v "i The c l a s s i f i c a t i o n o f c o u n t a b l e g r o u p s due t o C a m p b e l l ^ • i s a complete c l a s s i f i c a t i o n . I n o r d e r t o d e s c r i b e i t we w i l l need t h e f o l l o w i n g n o t i o n s . L e t F denote t h e set o f a l l row- f i n i t e m a t r i c e s o ver R . A l l m a t r i c e s c o n s i d e r e d w i l l be i n F A square m a t r i x w i t h an i n v e r s e w i l l be c a l l e d r e g u l a r . I f , A = (a. .) i s . an i n t e g r a l m a t r i x w i t h n columns and \ S - (&-^>&2>" I s a n o r ( 3 e r e d s e t o f n e l e m e n t s o f a group Q\ then Ag w i l l denote t h e o r d e r e d s e t ( h ^ , h 2 , . . . ) where h. = £ a. .g. . By a v e c t o r we mean a m a t r i x w i t h one rov; and by a v e c t o r module we mean an a d d i t i v e group o f v e c t o r s . I f A e F we w i l l denote by (A) t h e v e c t o r module g e n e r a t e d by t h e rows of A . , . Now l e t r be a f i n i t e . o r c o u n t a b l y i n f i n i t e c a r d i n a l . 10. Q v r i . l l denote th e module c o n s i s t i n g o f a l l v e c t o r s w i t h r c o o r d i n a t e s and' J w i l l denote t h e submodule o f Q c o n s i s t i n g of the i n t e g r a l v e c t o r s o f Q . I f M i s a submodule o f J and P i s an i n t e g r a l square m a t r i x o f o r d e r r t h e n t h e s e t o f i n t e g r a l v e c t o r s c w i t h cP e M i s a submodule o f J w h i c h v/e w i l l denote by M : P and c a l l t h e q u o t i e n t o f M by P . I f I i s t h e u n i t m a t r i x and P « ml (m e Z) , we w i l l w r i t e M : m f o r M : ml . Suppose t h a t G i s a c o u n t a b l e group and t h a t r ( G ) =» r . L e t g =a (g-L^gg/* • •) D e a n o r d e r e d b a s i s o f G g e n e r a t i n g t h e . b a s a l subgroup H . F o r m e Z , m > 0 , l e t f(m) d enote t h e set o f a l l i n t e g r a l v e c t o r s c w i t h eg d i v i s i b l e i n G by m . f(m) i s a submodule o f J . The f u n c t i o n f t h u s d e f i n e d on t h e p o s i t i v e i n t e g e r s i s c a l l e d t h e d i v i s i b i l i t y f u n c t i o n o f G w i t h r e s p e c t t o g , f o r i t c o m p l e t e l y d e s c r i b e s i n G ' t h e d i v i s i b i l i t y p r o p e r t i e s o f t h e e l e m e n t s o f H . Theorem 2.3 The f u n c t i o n f c o m p l e t e l y d e t e r m i n e s t h e group G t o w i t h i n i s o m o r p h i s m . Furthermore f(m) i s t h e i n t e r s e c t i o n o f a l l f ( q ) } where q ranges o v e r t h e p r i m e power f a c t o r s ofm . Hence f i s c o m p l e t e l y d e t e r m i n e d by i t s v a l u e s a t t h e p r i m e powers and the c o n d i t i o n f ( l ) - J . T h i s g i v e s f o r e a c h p e TT a sequence f ( p ° ) J 2 f ( p 2 ) "2 ... ?_ f ( p n ) 2 . . . . : The system o f t h e s e sequences, f o r a l l p e ir , i s c a l l e d t h e s d i v i s i b i l i t y system o f G w i t h r e s p e c t t o g . Groups w i t h a 11. common d i v i s i b i l i t y system are isomorphic. We also have that f ( p n + 1 ) • P - f ( p n ) f o r n " O A *2,... Nov; l e t A =. (A jA^,... ,An,...) be an i n f i n i t e sequence of submodules of J . We w i l l c a l l A a Dp-sequence (p e ir) i f A Q => J and A n + 1 : p « A / f o r n » 0,1,2,... . A system [A(p)] , containing p r e c i s e l y one Dp-sequence A(p) - (A (p), A x ( p ) , A 2(p),...)' f o r each p e TT , w i l l be c a l l e d a D-system. I t i s clear that /every d i v i s i b i l i t y system i s a D-system. We also have ' Theorem 2.k Every D-system i s a d i v i s i b i l i t y system of a s u i t able group G . • j A D-system/Which i s a d i v i s i b i l i t y system of a group G i s said to belong /to G . Thus.we have that any given D-system belongs to ^ a unique group. The determination of the d i v i s i b i l i t y system f o r the group G depended ^on the basis g . Thus d i s t i n c t D-systems may belong to isomorphic groups. To complete the c l a s s i f i c a t i o n we now look at the e f f e c t of a change of basis of G and deter mine conditions f o r two D-systems to be equivalent. An- ordered basis of G Is/ expressible i n the form Pg. , where P i s an / i n t e g r a l square matrix of order r . In.fact, Pg i s a basis of G i f f P i s regular. 'And so, i f ,[A(p)_] i s the d i v i s i b i l i t y system of G with respect to g j then the d i v i s i b i l i t y system of G with respect to Pg , where P. i s regular, is.,[A(p) : P], 12. Let g,g'/ be ordered bases of G generating the basal subgroups H,H! respectively. Then H n H ' i s a l s o a basal subgroup and we may select a basis g" of G i n H (1 H . For suitable regular/matrices P and S we. have Pg ~ g" =» Sg' . Suppose [A(p)] and [B(p)] are the d i v i s i b i l i t y systems of G with respect /to g,g' respectively. Then f o r a l l p e ir A(p) : P = B(p) : S . This r e l a t i o n provides the equivalence on the D-systems. We w i l l c a l l two D-systems [A(p) ] , [B(p) ] associated i f , f o r a l l p e TT and f o r suitable regular i n t e g r a l matrices P,S, independant of p , A(p) : P = B(p) : S . . » . The r e l a t i o n of association between D-system i s an equivalence. The r e s u l t i n g equivalence classes are c a l l e d D-types, The set of D-systems belonging to a given group Is a LVtype. Every D-type corresponds to some group and two groups are isomor phic i f f t h e i r D-types are the same. Hence the D-types, provide a complete c l a s s i f i c a t i o n of the countable groups of rank r where .r i s a f i n i t e or countably i n f i n i t e c a r d i n a l . The f i n a l c l a s s i f i c a t i o n f o r countable groups that we w i l l discuss here i s one f o r groups of f i n i t e rank due to Rotman. Let G and G be groups of f i n i t e rank r . Suppo i s a basis of G and y-j_*...,y r i s a basis of G ' such that r r , h ( E m.x , G ) m h ( E tn.y. , G ) f o r a l l primes p and a l l p i=.l 1 1 P i=»l • 1 x Integers . Define a mapping f : G - G ' as follows. Let f (\) = y.± (I = 1,2,... ,r) . Suppose O f x e G . Then there exists Integers m,m1,...mr with mx » E m±x± • W e c a n assume 1 5 . that m = p k for some p e ir and some k >_ 0 . Thus h (E m..x.) 2 V a n d "there e x i s t s a unique y G G * such that P ky' = Z m iy i . Set f(x) = y . f i s then a well defined isomor phism. The result i s : Theorem 2 . 5 I f G and G * are groups of f i n i t e rank r , then G s G * i f f there ex i s t bases x^, ...x r of G and y x , . . . , y r of G * with h p ( E ia xx ±, G ) = h ^ E m^ y ± , G') f o r a l l p e ir and; a l l integers cm_. 1 * r . Let Z r denote II. Z and l e t ,.. ,x be an ordered i - 1 x : basis of G . We define a height, function f : r x Z r -» N by f(p,m^,... ,m. p-) - hp(E " i ^ y G) . Such functions describe the groups, as theorem 2..5 i n d i c a t e s . But the d e s c r i p t i o n i s not complete as d i f f e r e n t "bases w i l l y i e l d d i f f e r e n t functions. Hence suppose that y x*...,y r i s another basis of G and suppose further that y x,...y r induces a function g : f x Z r N . Nov;, there exists a r a t i o n a l non-singular r x r matrix A '= (a. .) with y. • j a, .x. . Let n be the product of the denominators of a. . . Then ng. - E na ,x. . The c o e f f i c i e n t s i j 1 j i j J na . are a l l integers and we have i J g(p,nm1,.. .nmr) - f (p, 2 na^,... , 2 m x n a x r ) » f ( p / [m^,... ,mr ]nA) This r e l a t i o n i s an equivalence r e l a t i o n on the functions and any two ordered bases of G w i l l determine the same equivalence c l a s s 14. o f f u n c t i o n s . Hence, t h e e q u i v a l e n c e c l a s s i n an i n v a r i a n t o f G . Theorem 2.5 can now he r e s t a t e d a s ; Theorem 2.6 L e t G and G ' he groups o f f i n i t e r a n k r . Then G £ G ' i f f t h e y have t h e same e q u i v a l e n c e c l a s s o f h e i g h t f u n c t i o n s . Rotman t h e n d e t e r m i n e s w h i c h e q u i v a l e n c e c l a s s e s o f the f u n c t i o n s f : ir x Z r -» N a c t u a l l y c o r r e s p o n d t o gr o u p s o f rank r . We c o n c l u d e t h i s s e c t i o n w i t h a l o o k a t a c l a s s i f i c a t i o n f o r a r b i t r a r y groups due t o ErodHs. The a p p r o a c h here i s t h r o u g h t o r s i o n f r e e f a c t o r groups o f f r e e a b e l i a n g r o u p s . The c l a s s i f i c a t i o n r e q u i r e s some r e s u l t s on t h e s e g r o u p s . The main r e s u l t needed i s ; Lemma L e t F/H and F'/H' be I s o m o r p h i c t o r s i o n f r e e f a c t o r g r o u p s o f the f r e e a b e l i a n groups F and F' . Then t h e r e e x i s t s an isomorphism cp o f F onto F* w i t h H9 = H' i f f r ( H ) - r ( H * ) Here, as p r e v i o u s l y , m a t r i c e s w i l l be row f i n i t e m a t r i c e s over R . A l l m a t r i c e s w i l l , b e square m x m m a t r i c e s where m i s a c a r d i n a l . - A m a t r i x A i s c a l l e d r i g h t r e g u l a r I f t h e r e e x i s t s a m a t r i x A w i t h AA = I . We w i l l c a l l two m x m m a t r i c e s A and B e q u i v a l e n t i f t h e r e e x i s t r e g u l a r m a t r i c e s P and Q w i t h PAQ - B and b o t h Q and Q"1 a r e i n t e g r a l m a t r i c e s . Theorem 2.7 L e t m be any i n f i n i t e c a r d i n a l . Then t h e r e e x i s t s a one-to-one co r r e s p o n d e n c e between a l l groups o f c a r d i n a l i t y _< m (up t o isom o r p h i s m ) . and a l l r i g h t r e g u l a r m x m m a t r i c e s ( i f we do not make d i s t i n c t i o n between e q u i v a l e n t m a t r i c e s ) . 15. P r o o f : L e t F be a f r e e group of .rank m. Any group G o f c a r d i n a l i t y _< m i s i s o m o r p h i c t o a f a c t o r group o f F modulo a subgroup H o f rank m . L e t e a c h subgroup H o f F c o r r e s p o n d t o t h e f a c t o r group F / H . Then by t h e lemma and t h e above remark t h i s e s t a b l i s h e s a one-to-one c o r r e s p o n d e n c e between a l l groups o f c a r d i n a l i t y _< m (up t o Isomorphism) and a l l p u r e sub groups o f rank m o f F i f we do n o t make d i s t i n c t i o n between subgroups o f F w h i c h can be mapped o n t o e a c h o t h e r by automor phisms o f F . L e t F be a f u l l subgroup o f a r a t i o n a l v e c t o r space V . The correspondence S - S fl F I s one-to-one between sub- spaces S o f V and t h e p u r e subgroups o f F . We have t h a t r ( S ) = r ( s n F ) and t h a t fl F can be mapped o n t o S 2 n F by an automorphism o f F i f f V has an automorphism mapping S^ : onto S 2 and F onto i t s e l f . Thus our p r o b l e m I s e q u i v a l e n t t o the problem o f c l a s s i f y i n g a l l subspaces o f r a n k m o f V. un d e r the group of automorphisms o f V w h i c h map F o n t o i t s e l f . Nov; a subspace S o f V has r a n k m I f f V can be mapped o n t o S by an endomorphism w i t h a r i g h t i n v e r s e . I f • a r e endo- morphisms o f V. w i t h r i g h t I n v e r s e s t h e n « V-tg i f f t h e r e e x i s t s an automorphism cp o f V w i t h » £g '• An automorphism cp o f V maps 1 1 ^ onto Yl2 I f f t h e r e i s an automorphism w i t h "P-j^cp = - t 2 . The theorem, f o l l o w s f r o m t h e r e p r e s e n t a t i o n o f endomorphisms o f V by m x ni . / m a t r i c e s . I 1 6 . Now I f G i s a group o f c a r d i n a l i t y _< m , l e t G be r e p r e s e n t e d as F/H where F i s a f r e e group and r ( F ) = r ( H ) = m. t L e t b and b (xeA) be bas e s o f F and H r e s p e c t i v e l y . Then the m a t r i x A « (a ) d e f i n e d by b' «= s. a. b , \ e A , AM- X AM M a e R , i s t h e m a t r i x c o r r e s p o n d i n g t o G . Here, a s v / l t h t h e c l a s s i f i c a t i o n s o f the c o u n t a b l e g r o u p s , t h i s c l a s s i f i c a t i o n does n o t p r o v i d e a s a t i s f a c t o r y system o f i n v a r i a n t s because o f t h e problem o f d e t e r m i n i n g e q u i v a l e n t m a t r i c e s . . 3- I n d e c o m p o s a b i l i t y and D i r e c t Summation I n t h e p r e v i o u s s e c t i o n we mentioned a l l t h e c l a s s i f i c a t i o n s of the c o u n t a b l e and a r b i t r a r y g roups t h a t a r e known. rWe now mention and d i s c u s s b r i e f l y one f u r t h e r c l a s s i f i c a t i o n , namely the w e l l known c l a s s i f i c a t i o n o f t h e rank one groups by t h e i r t y p e s . F o r a d e s c r i p t i o n o f t h i s c h a r a c t e r i z a t i o n we r e f e r t p Fuchs [ l j ] . T h i s i s t h e o n l y s a t i s f a c t o r y c h a r a c t e r i z a t i o n f o r groups t h a t e x i s t . Not o n l y does i t c o m p l e t e l y d e s c r i b e t h e rank one grou p s , but i t a l s o c o m p l e t e l y d e s c r i b e s the. s t r u c t u r e o f c o m p l e t e l y decomposable groups ( g r o u p s w h i c h a r e d i r e c t sums of rank one groups) f o r Baer [2] has pr o v e d _^ """" Theorem L e t G be a c o m p l e t e l y decomposable group and suppose G - £^ (±) G^ a ^ r i © H n w h e r e G ^ a n d H n a r e r a n k one groups>~ ^ Then t h e r e e x i s t s a one-to-one c o r r e s p o n d e n c e between t h e summands G^ and t h e summands H^ s u c h t h a t c o r r e s p o n d i n g summands a r e \ i s o m o r p h i c . Hence any d e c o m p o s i t i o n o f a c o m p l e t e l y decomposable group I n t o a d i r e c t sum o f r a n k one; g r o u p s i s e s s e n t i a l l y u n i q u e . Unfortunately, the problem of determining those groups which are completely decomposable has not yet been solved. Some neces sary and s u f f i c i e n t conditions f o r a group to be completely decomposable are known but either these conditions are applicable only to r e s t r i c t e d classes of groups or the groups with the conditions have not been determined. For example, Rotman [35] has used his invariants f o r f i n i t e rank groups (see the previous section) to show that a group G of f i n i t e rank i s decomposable i f f G contains a basis ' x x,. ...,xr,y]L,.. .y g such that, f o r a l l p e ir and a l l m. ,m. e Z , h (£ m.x. + £ m.y.,G) . X j . p X X J J = min{h p(£ m^x^G), h ^ E n^y^G) } . This can obviously be used to derive a necessary and s u f f i c i e n t condition f o r the group G to be completely decomposable. However the problem of determining the groups with such a basis has not been solved. More of these conditions, along with further Information on completely decom posable groups, can be found i n Baer [2] , Kurosh [25]':-> Fuchs ,[1J] , and Wang I38] . We'-conclude our b r i e f d i s c u s s i o n of rank one groups and completely decomposable groups with a review of • some of the r e s u l t s on such groups, that have been published recently.' . . . Baer [2] has proved that i f G Is a completely decomposable group and the rank one summands of G a l l have the same type then every pure.subgroup of G i s a l s o completely • 1 decomposable. Prochazha [31] has generalized t h i s ' r e s u l t and proved that i f G = E © G i s a d i r e c t sum of rank one groups a < T : a v/hose types are inversely well ordered i n the/ natural p a r t i a l 1 8 . o r d e r o f t y p e s ( i . e . a < 3 < T i m p l i e s t y p e G^ < t y p e G Q) then any pure subgroup o f G i s c o m p l e t e l y decomposable. A c t u a l l y , i n p r o v i n g t h i s r e s u l t , P r o c h a z k a p r o v e s t h a t any p u r e subgroup o f G i s a d i r e c t summand o f G . Hence t h e r e s u l t i s s t r o n g e r t h a n i n d i c a t e d f o r any d i r e c t summand o f G i s com- p l e t e l y decomposable. Kovas [2J>] has shown t h a t I f G i s a group and H a subgroup o f G w i t h nG q,H f o r some, p o s i t i v e i n t e g e r n and i f e i t h e r G o r H i s a d i r e c t sum o f r a n k one grou p s whose t y p e s a r e i n v e r s e l y w e l l o r d e r e d I n t h e n a t u r a l p a r t i a l o r d e r o f t y p e s t h e n G s H . r . r H o m o l o g i c a l methods and t h e c o n c e p t o f r e g u l a r g r o u p s have been employed by H a r r i s o n [ 1 7 ] t o h e l p t h r o w l i g h t on t h e • problem o f d e t e r m i n i n g t h o s e groups w h i c h a r e c o m p l e t e l y decom- N •\ p o s a b l e and g a i n some i n s i g h t i n t o t h e number o f gr o u p s w h i c h a r e . ' " \ . not c o m p l e t e l y decomposable. L e t G be a group. F o r ' p e ir l e t \ f(p,G) denote t h e d i m e n s i o n o f G/pG a s a v e c t o r space o v e r t h e prime f i e l d o f c h a r a c t e r i s t i c p . We w i l l w r i t e f(.G) = ~ f p f ( p ' G ) . I f S- i s a pure subgroup o f . G , t h e n f ( S ) . f(G/S) a f ( G ) ( i . e . f ( p , S ) + f (p,G/S) = f ( p , G ) f o r a l l p ) . IT r We v / i l l a l s o w r i t e n(G) = }* p where r r ( G ) I f H i s any subgroup o f G n(H)' . n(G/H)"= n(G) . Fo r p e ir l e t A_. be t h e subgroup o f R w i t h IT denominators o f powers o f p . We w i l l w r i t e j £(p,G) « r(Hon(A ,G)) and e(G) ="n~pe(P^G) . Then i f j H i s any tr Ir \ \ subgroup o f G e(G) _< e(H).e(G/H) ./ These t h r e e f u n c t i o n s a r e used to define regular groups. For any group G, f(G) . e(G)_< n(G) T/7e w i l l c a l l a group G regular i f r(G) i s f i n i t e and f(G) . e(G) = n(G) . I f G i s a regular group and S i s a pure subgroup of G 3 then both . S and G/S are regular. A l l rank one groups and a l l f i n i t e d i r e c t sums of rank one groups are regular. Harrisons r e s u l t s are contained i n two theorems which are; 1. A group G which Is d i v i s i b l e f o r a l l but a f i n i t e number of primes i s a d i r e c t sum of a f i n i t e number of rank one groups ' - i f f i t -is regular. I f S i s a pure subgroup of a group\ G - x which i s a f i n i t e d i r e c t sum of rank one groups a l l o f which c : \ \ are d i v i s i b l e f o r a l l but a f i n i t e number of primes then both^ S and G/S are also d i r e c t sums of rank one groups. 2. Let G and H be groups of f i n i t e rank such that G i s d i v i s i b l e for a l l but a f i n i t e number of primes. Then the « number of non-isomorphic groups . K which have a subgroup H i isomorphic to H and with f a c t or group K/H isomorphic to G i s either one or the c a r d i n a l i t y of the continuum depending on whether or not n ( G ) / f (G) i s r e l a t i v e l y prime to f(H) ( i . e . whether or not ei t h e r r(G) - f(p,G) = 0 or f(p,H) » 0 f o r a l l p e TT ) . \ \ These r e s u l t s show the immense number of groups which exist even f o r small ranks and that only r e l a t i v e l y few of them are completely decomposable. For example the number of rank two groups formed by putt.ing A and/A (see above) together, where. q / P ' • . I • P k P J ^ E t i ' s power of the continuum but only one of them i s completely decomposable. 1 20. Theorem 3.1 shows t h a t d i r e c t sums o f r a n k one groups p r e s e n t no i r r e g u l a r i t i e s f o r ' t w o d i f f e r e n t d i r e c t summations o f r a n k one groups y i e l d n o n - i s o m o r p h i c g r o u p s . I t i s n a t u r a l l y t o be hoped t h a t t h e p r o p e r t y o f theorem 3.1 w i l l c a r r y o v e r t o d i r e c t sums of a r b i t r a r y indecomposable g r o u p s . U n f o r t u n a t e l y , t h i s does n o t happen. We w i l l now demonstrate t h a i , i n g e n e r a l , d i r e c t . summations behave i n a" v e r y e r r a t i c manner. F o r i n s t a n c e , C o r n e r [ 8 ] , u s i n g one o f h i s own r e s u l t s , [7] > has c o n s t r u c t e d , f o r any p o s i t i v e i n t e g e r r , an example o f a c o u n t a b l e , reduced;- group /G . w i t h t h e p r o p e r t y t h a t £ G a £ G i f f m s n(mod r ) . m n Here m and n a r e p o s t i v e i n t e g e r s and £ G i n d i c a t e s t h e n d i r e c t sum o f n copies,'of G . I n p a r t i c u l a r , t h e r e t h e n e x i s t s a. group G t h a t i s i s o m o r p h i c t o t h e d i r e c t sum o f r + 1 c o p i e s o f i t s e l f b u t n o t t o t h e d i r e c t sum o f s c o p i e s o f i t s e l f f o r a l l 1 < s < r +/l . ' <• The main problem i n t h i s c o n n e c t i o n i s p r e s e n t e d by d i r e c t sums of indecomposable g r o u p s , n o t a l l o f w h i c h a r e r a n k one groups. I t i s / p o s s i b l e f o r two d i f f e r e n t ( i n t h e sense o f theorem 3-1) ' d i r e c t sums o f i n d e c o m p o s a b l e g r o u p s t o be d i r e c t d e c o m p o s i t i o n s 'of.the same group. T h i s - p r o b l e m i s f u r t h e r . c o m p l i c a t e d by t h e f a c t t h a t b e s i d e s t h e r a n k one groups w h i c h are o f coursed/indecomposable, v a s t numbers o f i n d e c o m p o s a b l e groups a r e knovm t o . e x i s t . We w i l l f i r s t d i s c u s s some examples of indecomposable groups and t h e n some examples t o i n d i c a t e t h e f a i l u r e o f theorem 3.1 i n t h e g e n e r a l s e n s e . Rotman [35] has used h i s i n v a r i a n t s ( s e e p r e v i o u s s e c t i o n ) t o c o n s t r u c t examples o f i n d e c o m p o s a b l e g r o u p s o f any f i n i t e 21. rank. The pure subgroups o f Z^ y i e l d examples o f Indecompos a b l e groups o f any r a n k up t o and i n c l u d i n g tf ( t h e power o f the continuum), de Groot'and de V r i e s [16] have c o n s t r u c t e d examples o f rank m f o r .tf' < m < . i n d e c o m p o s a b l e g r o u p s ' J¥ o - - of any rank _< 2 a r e p r e s e n t e d i n Fuchs [ 1 3 3 . The b e s t p o s s i b l e r e s u l t however I s t h e example c o n s t r u c t e d by Fuchs . [ 1 4 ] We w i l l l o o k a t t h i s example i n some d e t a i l . Fuchs p r o v e s Theorem 3.2 F o r e v e r y i n f i n i t e c a r d i n a l l l i i " t h e r e e x i s t s a r i g i d system o f groups o f power m . By a r i g i d system o f groups we mean a s e t o f g r o u p s , G (x^A) i w i t h t h e p r o p e r t i e s - X ( i ) I f \ ^ n t h e n Ebn(G ,G ) » 0 f o r a l l \, n e \ . K n - - (ii) F o r e v e r y endomorphism >cp" o f G^ t h e r e e x i s t s r ^ e - R . such t h a t gcp « r g f o r a l l g e G . , Groups i n any r i g i d ^ . • \ system a r e indecomposable and p a i r w i s e n o n - i s o m o r p h i c . We w i l i \ ^ use t h e f o l l o w i n g n o t a t i o n i n t h e c o n s t r u c t i o n . I f t i s a type^ and G i s a group, r e c a l l t h a t we w r i t e G t a [g e G | t(g,Q) • L e t t Q » [(»,0,0,...) ]• and '•t 1 = [ ( o , « , o , o , . . . ) ] . 1 Lemma I f f o r some c a r d i n a l m we have a system o f g r o u p s G (x£A) s a t i s f y i n g - ' | '•• . ' - - . : -.i • 1 • | G | = m • • ( . " ' . o 1 x' ! 2 0 I M - 2m , / 4 • for every X e A/ elements , g^/g\ e G>v ^ 2 G x ' which are V G X X'- X lndependant mod/G , can be selected such that \ G* = [G , 2_c°g/, 2"°°g } ( i . e . the group generated by the d i r e c t X X X X sum of G ar/d the elements of the form 2" ng , 2 ~ n S . j n«l,2, X I \ x i has the property: i f cp i s - a homomorphism of •' G;' Into G then / x ^ either cp «yO or .\ = n and there..exists r £ R with gcp-» rg for a l l g/e G^ ... Then, for any/ n with m < n ..2^ there e x i s t s a system of / groups H (aeA) which also has properties 1^ - 4^ Note that any system s a t i s f y i n g properties 1^ - 4-^ i s * a r i g i d system by property 4^ ,. The construction of the^ K& proceeds as follows; l e t A (aeA) be a c o l l e c t i o n of subsets of a i A, each of power n , such that x o e \ f o r a f i x e d X Q and every a and ^ H A^ implies a a b. There e x i s t s 2 n sets A with these properties and hence we may assume A has power 2 n . For every A l e t H - { £ G . ••2"eD(g, +g'),...}'> a a x e x X Q X (x' f \ q J X .e A a , g^ e G^) . As the lemma indicates the construction w i l l y i e l d a r i g i d system with add i t i o n a l properties to ( i ) and ( i i ) . The^ next step i n the construction i s to construct a system of 2: groups of power ,V i • Let G^ x e A ) be rank one groups whose 23. types are pairwise Incomparable and incomparable with the types ; and t.. Q- 1 t ^ . a n d t n . We may assume the Index set has power A j / , . V/e can form 2* A subsets A a(aeA) of A such that |Aa| - y\\ > a fixed x_ belongs to a l l A and A c A implies, a <=» b . a a —" Select e G^ , $ 3G^ and, f o r every A & , define H = { S G , 3"" (g . + g ' ) , . . - 3 U* *> X Q, \'.€ A , g. e G ) . a XeAQ x xo x 0 a x X g + g' w i l l have type t. In H and the system H a(aeA) has Xo x properties 1 - 2K The f i n a l step i s to construct a system of groups w i t h properties 1 - lK for a l i m i t cardinal n . Let c 1 0 0 ~ )V-n =» nx. ,nu, ... ,m ,... b e t h e sequence of a l l cardinals greater than »V and less than n . Suppose that for each -nx, there o , . , . pt exists a system of groups s a t i s f y i n g 1-w - 4 . With the G as In the previous step, a system K i s constructed X x for n , such that every K . a r i s e s as the union of a sequence (1) (2) (a) G x c H a c c _£ ... s H c £ ••• • Different sequences may produce isomorphic K . We consider only the non-isomorphic K that r e s u l t . I t can then be proved that the K have property 4 Q and that there are at l e a s t n non-isomorphic groups K . Then, applying the procedure of the lemma we ob- N t a i n 211 groups of power n s a t i s f y i n g the conditions 1 Q - 4 Q . 24. This completes the construction and the proof of theorem 3.2. V/e remark that t h i s result, can be used to answer problems 20, 21 and 46 i n Fuchs book [13] . The f i r s t example demonstrating that d i f f e r e n t d i r e c t sums of indecomposable groups could y i e l d isomorphic-.grbups was due to Jonsson [19] who discovered a rank 3 group that could be decomposed into a d i r e c t sum of indecomposable groups i n t w c T ^ d i f f e r e n t ways. Jesmanowicz', [18] presented another example in answer to a problem posed by Fuchs [13* problem 22] . Fuchs asked: I f ^(1=1,2,3,4) are p o s i t i v e integers with . r1 + r2 ~ r ^ + r ^ and r^ ={= r ^ , r ^ J» ri± t does there-- e x i s t inde composable groups 0^1=1,2,3,4) with r(G i) ^ r ^ i - l , 2,3,4) and ® G 2 s @ . Jesmanowicz answered t h i s problem aff i r m a t i v e l y and also proved a s i m i l a r r e s u l t f o r three d i r e c t — summands. The strongest example i n t h i s connection i s ' due to Corner [6] who generalized Fuch's problem and answered i t aff i r m a t i v e l y . Theorem 3 . 3 Let N,k be p o s i t i v e integers with N _> k . Then there exists a group G of rank N such that for. any p a r t i t i o n , / N = r n + r 0 +...+ r, ., of N i n t o vk- positi v e integers there ex i s t s indecomposable subgroups G1,...,Gk of G s a t i s f y i n g ''./.•' • (i) r(G i) = r ^ i - l , . . . , ^ (ii) G = ^ © G . 1=1 1 25. Proof: Let n » N - k I f n = 0 take G to be free with r(G) «» k = N . Hence we may assume n _> 1 . Let Pa Pi>"«.»P n> <l 1*««»*q n' D e d i s t i n c t primes and l e t N u 1,...,u k, x 1,...,x n be a basis of R . Let G be the group generated by {p*"^, P j ' V j * q j 1 ( u l + x j ^ C1^1^* ^ J i , 1 1 * m2P) • Then G i s a subgroup of R and r(G) » N . / Nov/ l e t r ^ +...*+ r k = N be a p a r t i t i o n of N i n t o k positive integers. Set SQ = 0 and s^ « r ^ +...+ r^ - i (i»l, ...,k). I f v/e write ^2."''%. " then (^,...,0^) = 1 and there exists t, , . . . , t e Z such that E 1 1 t. Q, =1 . Let 1 n / i - l 1 1 7 a. = E "t.Q. . Then ' E^ a. =1 and a. = 1 (mod q.) S I - 1 < J - S I 3 J I»l 1 . . 0 i f s. T < j < s.; a. s 0 (mod q.) otherwise. We now define i - l 1 I /* j b.,...,blr 6 R N by: u l = alV/+ a 2 b 2 + a 5 b 5 + + a k b k U 2 = " b l f b 2 u, o -b- + "b, 0 3 1 . 3 u k = " b l • + b k • These equations/can be solved to give b,,... ,b, as i n t e g r a l . / K • i l i n e a r combinations of ••u^,...,uk * j 26. For i = l,...,k , jlet G± be the group generated by {p"\, q j 1 ^ i + x j ) 5 ( s i _ i < J < s x, m > 0) . These G^ are the/required indecomposable groups. Corner Also proves a similar result for the countable rank care, namely; Theorem 5.4 There "exists a group G of countable rank such that for any sequence r x , r 2 , . . . of positive integers, i n f i n i t e l y many of which are greater than one, there exists indecomposable subgroups G^ of G satisfying ( i ) r(G.) » r. (i-1, 2 , . . . ) (iL) G . E @ G . i 1 4. Quasi-isomorphism and Quasi-endomorphisms. . ; The examples of section 3 indicate that a large number of groups exist which do not have unique decompositions into a sum of indecomposable groups. This i s because the condition of uniqueness up to isomorphism i s too strong. If we replace Iso morphism by the weaker condition of quasi-isomorphism and inde- pomposability by the corresponding notion of strong indecomposabil- i t y , we retain the essential uniqueness of decompositions of groups of f i n i t e rank. ' In this section we define and discuss the notion of quasi-isomorphism and state a theorem analogous to i theorem 3.1 . We w i l l also define and establish the importance of the notions of quasi-endomorphisms and the ring of quasi- endomorphisms of a group. • • • ' / ' . • 27. The definition of quasi-isomorphism i s due to Jonsson [20] and.was originally given for groups i n general. Definition 4.1 Let G and H he groups. (i) G and H are said to be quasi-isotnorphic i f f each i s isomorphic to a subgroup of the other. We w i l l write G £ H •;'/ ..." (i i ) G i s said to be quasi-contained i n H i f f G's Gyn-H-; We w i l l write G c H . (iii) G and H are said to be quasi-equal i f f G £ H and H c_. G •. We w i l l write G = H . (iv) G i s said to be quasi-decomposable i f f there exist non zero independent groups G^ and Gg , such that G & G^ Q G g . (v) G i s said to be strongly indecomposable i f f i t i s not quasi- decomposable. ... ' • Other formulations of these notions,-hetter suited to. the case of torsion free-abelian groups are as follows. Let G and H be torsion free abelian groups. Then \ x (i) G s H i f there exists subgroups . G ' C G, H* C H and positive • f i i r \ integers tn,n .such that G £ H , nG e G , and mH c H . \ i • \ /(Li) G o H i f , for some positive integer n, nG c H \ \ If G and H are torsion free abelian groups of f i n i t e rank then ' i ' / ( i ) " - G s H i f there exists subgroups G c G , H 'cz H such I t I ! that G £ H and G and H >. have f i n i t e index,in G and H respectively. Also G k H i f there exists a subgroup H' C H such that H* has f i n i t e index i n H and G = H* . 2b. (i) - (v) are Jonsson's o r i g i n a l d e f i n i t i o n s . As defined by Jonsson a • i s an equivalence r e l a t i o n on the class of a l l groups, c i n general i s not t r a n s i t i v e over the class of a l l subgroups of a group G and hence i need not be an equivalence r e l a t i o n , (i) ' i s from Beaumont and Pierce [3] . ( i i ) i s from Ried [33] and ( i ) " from Beaumont and Pierce [4] . ( i ) " can be seen by applying Theorem 2.4 of Jonsson [20] . Both £ and i are equivalence r e l a t i o n s on the class of a l l torsion free abelian groups. The d e f i n i t i o n s of quasi-isomorphism was, as we stated above introduced by Jonsson to provide the following theorem, analogous to theorem 3.1 • Theorem 4. 2 Let G^Gg,... ,Gm, H^Hg,... ,H n be strongly i n d e r composable groups of f i n i t e rank such that if 1 1 @ G. a ^ ( ^ H . 1*1 1 j«l J "" Then m.« n and there^exlsts a permutation, cp of [1,2,. ..,m) such that G^ = Hcp(i) •••*ro) • Actually a stronger r e s u l t than t h i s i s possible i f we make use of the ring of quasi-endomorphisms of a group. We w i l l give t h i s presently. As a re s u l t of theorem 4.2 , quasi-isomorphism has come to play an important role i n the study of groups. Some properties of groups that are invariant under quasi-isomorphism have, been found and i n the case of rank two groups a complete set of quasi- isomorphism invariants have been found by Beaumont and Pierce [4] We w i l l now review some r e s u l t s on quasi-isomorphism. The i n v a r i - j for the rank two groups w i l l be discussed i n another section. 2 9 . Beaumont and Pierce [?] have proved that i f G s II and G* & H* then G © G ' a H@ H* . It has also been proved here that i f G and H are groups then the following conditions: are equivalent (i) G k H (i i ) there exists subgroups G* c G , H* C H and a positive i t t integer n such that G £ H , H « G 3 n G c G and n H c H .. (iii) There exists a subgroup G ' C G and a positive integer n such that G s H and nG c G . If G and H are quasi-isomorphic groups then. r(G) -'r(H) and T(G) - T(H) ( [ 4 ] ) . Hence rank and type sets are quasi-isomorphic invariants.. Koehler [22] has proved some properties of quasi- isomorphic groups of f i n i t e rank. / If G and H are groups of f i n i t e rank then'we can consider them as both being subgroups of R N for some n . Then the following conditions are equivalent. (i) G k H ' ( i i ) there.exists a subgroup H of H and a monomorphism cp from H' to G such that G c cp(Hf) and H c H* . ( i i i ) there exists a monomorphism cp from H to G such that G c cp(H) c G . . . ' . . t i , (iv) there' exists a subgroup G of G such that H £ G « G . (v) there exists non-singular linear transformations X1,X2 of R N such that ^(G) c H and \ 2 ( H ) . G G . Another r e s u l t proved by Koehleruses the notation: G denotes the minimal d i v i s i b l e group containing G . Now l e t G and H be quasi-isomorphic subgroups of R n . Then (I) G. S H f o r a l l types t . ( i i ) there exists a non-singular l i n e a r transformation L of R n such that I i ( l t ) = "G"t f o r a l l types t . ( i i i ) I f G = H then G t «= H t ; G t = H^ . f o r a l l types t . Note that "5^ , "H^ are sub spaces of R n . We conclude our discussion of quasi-isomorphism with an alternate d e f i n i t i o n proposed by Walker [J>J] . Walker considers the quotient category G/R where G i s the category of a l l abelian groups and is i s the class of a l l bounded abelian groups. two abelian groups G and H are quasi-isomorphic I f there e x i s t s i • t isomorphic subgroups G and H of G and H re s p e c t i v e l y With G/G1 and H/H* e f t . I f G and H are t o r s i o n free groups then t h i s i s equivalent to each being isomorphic to a subgroup^of the other with bounded quotients. Hence two t o r s i o n f r e e groups are quasi- isomorphic i f f they are isomorphic In Q/H . Furthermore \ ~ v quasi-decomposition, quasi-endomorphisms (see d e f i n i t i o n 4.4) etc. c . . • • • \ become decompositions, endomorphisms etc. i n the quotient cate- \ gory G/& . Hence quasi-decomposition theory of t o r s i o n f r e e groups i n G i s equivalent to decomposition theory pf t o r s i o n free groups i n G/B . Since quasi-isomorphism and quasi-decomposi-, • i . t i o n theory are of p r i n c i p a l value only i n the study jof t o r s i o n free groups, Walker submits that the proper d e f i n i t i o n of quasi- isomorphism should be: 31. D e f i n i t i o n 4.3 Two groups G and H are said to be quasi-isomor phic i f f they are isomorphic i n Q/» • This d e f i n i t i o n makes av a i l a b l e f o r a p p l i c a t i o n the homological algebra of G/B- and category theory i n general. We now discuss quasi-endomorphisms. Defin i t i o n 4 .4 Let G be a f u l l subgroup of a r a t i o n a l vector space V . Let L(V) denote the r i n g of l i n e a r transformations of V . We define E(G) = [X e L(V) | G\ c G} . I f \ e E(G) we c a l l x a quasi-endomorphism of G . I f \ e L(V) i s non^~>* -1 , X singular and \,\ e E(G) then we c a l l \ a qua si-automorphism of G . Hence x i s a qua si-automorphism of G i f f Gx « G . \ ^ E(G) i s the r a t i o n a l algebra generated by the endomorphisms of G i n L(V) and, i n p a r t i c u l a r , i s a r i n g and the quasi-automor- phisms are i t s units. / E(G) plays an important part i n the quasi-decomposition theory of G . This has been demonstrated i n two papers by Reid [33], [34] . We w i l l e s t a b l i s h the importance of E(G) by reviewing the pertinent r e s u l t s from these papers. I t i s obvious that G « G 1 © G 2 where G 1 = Ve H G = Ge arid • ; G 2 = V(l-e) H G « G(l-e) where e i s an id.empotent of E(G) . Conversely i f G L @ G g then there e x i s t s a unique idempotent e with G 1 = Ge and G 2 = G(l-e) ./Hence, we have that G i s ' /" I strongly indecomposable i f f E(G) contains no proper (4=0,1) . • /• . i • ' ' idempotents. l: 22. Now l e t G = Z n @ G . be a f i n i t e quasi-de compos i t ion i = l of G . Set H = 2 n © G. . Then G = H and E(G) = E(H) . i=l 1 Hence the projections 4^(1=1,-••,n) defined by the decomposition of ' H belong to E(G) .'. Furthermore they are mutually orthogonal idempotents and t h e i r sum i s the i d e n t i t y of E(G) . Also Gt, = G., (i=l,« • • ,n) and G= ^ © G t . . Thus any f i n i t e 1 1 • i = l 1 - quasi-decomposition of G i s s quasi-equal to one of the form G = E n (£) Gi. where l, are mutually orthogonal idempotents 1=1 1 1 whose sum i s the i d e n t i t y of E(G) . A quasi-decomposition of this form i s said to be normalized. Next, i f l-^'' ' ' are mutually orthogonal non-zero idempotents whose sum i s the identity of E(G) then E(G) has a decomposition " E(G) = 2 n © ^.E(G)" into a d i r e c t sum of r i g h t i d e a l s . 1=1 1 Theorem 4.5 The correspondence G = E n + Gi. - E(G) = Z n + l.E(G) 1=1 1 1=1 1 . i s one-to-one between normalized f i n i t e quasi-decompositions of G and f i n i t e decompositions of the E(G)-module E(G) . Also E(Gt x) l]E{G)l^ (i=l,-.«,n) and Gl± i s strongly indecomposable i f f £^E(G)• i s an indecomposable E(G)-module. If e and f are any idempotents of E(G) then Ge = Gf / • • . i f f eE(G) ; and fE(G) are isomorphic E(G)-modules; 33. Theorem 4 . 5 shows that the quasi-decomposition theory of G Is equivalent to the decomposition theory of E(G) as a (right) module over i t s e l f . As a r e s u l t of t h i s theorem we have the following generalization of theorem 4 . 2 : Theorem 4 .6 Let E(G) have descending chain condition on r i g h t i d e a l s . Then any quasi-decomposition of G has only f i n i t e l y many summands. I f zf11 (j) G.'-. G = s 1 1 © H. , where a l l G. ,H. . i - l • j-1 J J . are strongly indecomposable, then m - n and f o r some permuta t i o n cp of {1, ...,m], G i & Hcp'(i) ( i ^ l j ' * * * 1 1 1 ) • I f G has f i n i t e rank then E(G) i s a f i n i t e dimensional r a t i o n a l algebra and. so has descending chain condition on r i g h t i d e a l s . There also exists groups of i n f i n i t e rank with E(G) s a t i s f y i n g the descending chain condition. Hence theorem 4 . 6 i s a stronger r e s u l t than theorem 4 . 2 . n , , — A ring E with r a d i c a l N i s said to be completely primary i f E/N i s a d i v i s i o n r i n g . I t i s said to be semi- primary i f E/N has descending chain condition on right..ideals. Nov; suppose that E(G) has descending chain condition on r i g h t i d e a l s . Then the following are true, as Reid [34] has proven. (i) I f \ e E(G) then there e x i s t s a quasi-decomposition i G * G 2 such that, x induces a quasi^automorphism on G-j^ and a nilpotent quasi-endomorphism on G 2 ., ( i i ) G i s strongly indecomposable i f f E(G) i s completely primary. ' i- • ' \ I t i s also true that i f E(G) i s semi-primary-with n i l r a d i c a l , then G has a quasi-decomposition i n t o a f i n i t e number of strongly indecomposable summands whose 34 quasi-endomorphism rings are completely primary* Any two such quasi-decompositions of G are equivalent i n the sense of theorem 4.6 . We conclude this section by remarking that E(G) i s a quasi-isomorphism invariant. In fact If G <= H then E(G) = E(H) and i f G k H then E(G) « E(H) . . 5. Rank Two Groups Considerable effort has been devoted to the study of rank two groups. The reason for this i s two fold; to help develop a complete picture of rank two groups; and, since rank two groups are relatively easy to work with and exhibit, much of the pathology of groups of higher rank, to provide a basis for conjectures concerning groups"of arbitrary f i n i t e rank. Beaumont and Pierce [4] have classified the rank two groups up to quasi- isomorphism and used their Invariants to determine conditions for rank two groups to be quasi-decomposable and for^quasi-isomorr phic rank two groups to. be Isomorphic. They have, also employed their invariants in determing E(.G) for rank two groups and .in---, determing both necessary and sufficient conditions for a set of types to be the type set of a rank two group. Reid [3^] has \ provided another approach to determing E(G) for rank two groups and Dubois has devoted two papers, [10] and [11] , 1 to determing type sets of rank two groups. This section w i l l be devoted to reviewing and discussing some of the results of Beaumont and Pierce. The others w i l l : be presented In the follow- ing sections. 25. We start by outlining the development of the quasi- isomorphism invariants for rank two group of Beaumont and Pierce. We w i l l employ the following notation. (Gjx^Xg) w i l l be used to denote a f u l l subgroup,! G , of a two-dimensional rational vector space, V , with x x*x 2 forming a basis of G . If x e V then h p(x) w i l l denote sup {k| ( c/p k)x e G, (c,p) - 1, ceZ If x G G then h G(x) = h (x.G) . w i l l denote the ordinary p p x p logarithmic p-adic valuation on R and . Given (G;x1,x2) , i f and Hg are the pure subgroups of G generated by x^ and respectively, then G/(Hj@H2) i s either 0 or a torsion group. Hence we can make the following definition. Definition 5.1 Let Z be the characteristic (i . e . a function ---. from TT to N) satisfying G/(H,@Hp). £ E Z(p S( p)) . We w i l l • " . ' pGTT :V'.'.:y write (Gjx-^Xg) - E • • Now define A « {(a,p) | a,0 G TT z( p ) ] and l e t 6 pGTT be a characteristic. A pair (a,B) G A i s said to be G-equivalent to (a.i:0 ) 6 A i f (i) h p(a(p)) . hp(a\(p)) , hpO(p)) - l^ - ( P^P)) for a l l p . ( i i ) h p(a(p )3 ,(p) - af'(p)ft(p))'-> «(p) + h p(a(p)) + h p O(p)) . . . . . . . . . . . . j for a l l p I i We w i l l write (ct,0) ~ 0 (a',p') . ! 3 6 . ~ e I s an e q u i v a l e n c e r e l a t i o n on A . F u r t h e r m o r e i f (a,p) e A , (a',B') e A and i f f o r a l l . p e ir ( a ( p ) , B ( p ) ) and ( a ' (p) ,s ' (p)) s a t i s f y t h e f o l l o w i n g c o n d i t i o n s w i t h i ^ i ^ j ) a p a i r o f p - a d i c numbers. (a) hpCo^ ) « h ^ ( x 2 ) - u and h p ( P 1 ) - ^(x^ •= -.v (b) I f 0 < k C z(p) and i f m,n a r e i n t e g e r s w i t h h p(m) « u , .hp(n) «= v t h e n p ~ ( k + u + v ) (nxj+nxg) e G i f f h p ( m p 1 - n a 1 ) j> k + u + v , t h e n (a,S) ~ 2 (a',B') . T h i s r e s u l t l e a d s t o D e f i n i t i o n 5.2 L e t ( G j x - ^ X g ) -» £ and l e t (a,6) e A w i t h ( a ( p ) , B ( p ) ) s a t i s f y i n g c o n d i t i o n s (a) and (b) above f o r a l l p We d e f i n e X <= [ ( a , B ) ] . t o be t h e E -equivalence; c l a s s i n ; A c o n t a i n i n g (a, 8 ) . We c a l l ( E , X ) t h e p a i r o f i n v a r i a n t s d e t e r m i n e d by ; ( G ; x 1 3 x 2 ) and w r i t e : ( G j x ^ X g ) • ( z , X ) . • . Now assume t h a t 5^(p), 4= 0 a n d "that a ( p ) <= 0 . Then* hp(a(p)) «= oo and so a l s o h p ( x 2 ) = » . But t h i s i m p l i e s t h a t E(p) = 0 . Hence a ( p ) j= 0 . A l s o B(p) |= 0 . Thus we have i f E(P) r" 0 t h e n b o t h a ( p ) and p(p) a r e n o n - z e r o and so f o r \ . . . a l l ( a ' , 8 1 ) € X i f J£(p) ^ 0 t h e n a' (p) f 0 and p'(p) f- 0 . i t i s t h e n p o s s i b l e t o p r o v e j • | Theorem 5*3 . ! i ( i ) L e t x 1 , x 2 be an i n d e p e n d e n t p a i r i n V _ _ and j ( E , X ) a p a i r c o n s i s t i n g o f a c h a r a c t e r i s t i c £ and a £-equivalence 37. class X such that for a l l p e ir and (a,8) e X , i f E(p) ¥ 0 t n e n a(p) 1= 0 A N D P ( P ) ¥ 0 • T n e n t n e r e exists a f u l l subgroup G of V , ' such that x ^ X g e G and ( G;x 1,x 2) T * ( E , X ) ( i i ) Let ( G J X 1 , X 2 ) - ( E , X ) and (Hjy^yg) - (£,X) . Then the non-singular linear transformation cp of V taking x^ to y 1 and x g to y 2 satisfies cp(G) - m H i f f E = E and X = X . From now on a pair ( z , X ) w i l l be a pair such as i n theorem 5 . 3 (i) . As a result of this theorem, the correspondence between the f u l l subgroups ( G;x 1,x 2) of V and the pairs ( E , X ) i s one-to-one, for an independent pair x^,x 2 i n V . The next step i n the development i s to determine condi tions, i n terms of the Invariants, for groups to be quasi-isomor phic. If cp^and "Pg are functions from ir to Z u {»} we define ^1 < ^2 i f ~ ^ 2 ^ for almost a l l p , including a l l p with cp1(p) = «» . cf^ ~ cp 2 i f cpx < cp2 and <P2 < cpx . Now for a € TT we define a function K(a) p€ir from ir to Z U {»} by K(a) (p) = hp(a(p)) . We use this function to define an equivalence on the set of invaraints ( E , X ) . Definition 5.4 Let ( G J X ^ ) - ( E , X ) and (H;y1,y2) - ( E , ^ ) with (a,p) e X , (c7,B) e Y . We define ( E , X) ~ (E,7) i f : X • ~ " ': - • _ ' ' V '' ' \ x K(a) ~ K(a)' , K(B) ~ .K(B) and E + K(a) .+ K(s) < K(aB'-fef3) . \ ' ' \ If G and H are f u l l subgroups of V and x^,x 2 i s ; \ 38. an independent pair in GO H and i f (Gjx^Xg) -* (E,X) , (H;x 1,x 2) - (IT,X) , then G = H i f f (E,X) ~ ; (iT,X) . Making use of this result and theorem 5.3 we have • , Theorem 5.5 Let G and H be f u l l subgroups of V , xi>x2 an independent pair in G and l e t (GjX-^Xg) - (Z,X) . Then G = H i f f there is an independent pair .y x*y 2 i n H with (H;y 1,y 2) - (E~,X) such that (E,X) Z (?,X) . If we have (Gjx ^ X g ) - (E,X) then the pair (£,X) depends upon the basis xi>X2 o f G • T h e f i n a l s t e P i n the development of the invariants is to determine the effect on (E,X) of a change in basis; i.e. i f ( G ^ j X g ) -* (E,X) and y, ,y« is another independent pair i n G to determine (^,X) I t - . ' / where (G;y 1,y 2) "* ( E ~ , x ). Aong with this goes the problem of strengthening theorem 5*5 by determining s t r i c t l y In terms of the invariants, a condition for two groups to be quasi-isomorphi regardless of the basis chosen. A condition for them to be < isomorphic has not been found. Let xi»x2 > ^1*^2 fee ^n^eP 6 1 1^ 1 1* pairs i n G . Then there exists T\»i2 9' S X ' S 2 € R such that y^ = r^x^ +'-vsix-g \ • (i=l,2) . Suppose (G;x x,x 2) - (2~,X) with (a,p)eX , (-a,0)eX • y ' .-• \ It is possible to determine £(p), a(p), |3(p) for almost a l l primes p . Let u(p) = h^ , G /(y 2) , v(p) = h^y^) .; Then we define ' 39. L(p) - 0 i f h (Sla.(p) - I ^ B C P ) ) / h (s 2a(p) - r 2S(p)) > E(p) '+ hp(a(p)) + hp(6(p)) Z(p) + hp(a(p)) + hpP(p)) - (u(p) + v(p),.). otherwise / • a(p) = 0 i f u(p) = p u ( p ) i f u(p) ,/» and S(p) « 0 ' = -s po(p) + r 2 e{p) i f 2(p) > 0 8( P) = ;o i f v( P) «/«. = p v ( p ) i f v(p) < » and 2(p) « 0 . - S ; La(p) - r/e(p) i f £(p) > 0 . Then, i f ' X = [ ( ^ B ) ] i s the 2 equivalence c l a s s containing (a,p), ("sVx") ~y/(E*x) • The f i n a l r e s u l t , which demonstrates that t h e pairs (E',X) can be used to provide a complete quasi- / • : • . isomorphism-classification f o r the rank two groups, i s ; Theorem 5.6 . Let (Gjx^Xg) - (E,X) and (Hj y 1 , y 2 ) - (2,3C) . Then G a H i f f . . ^ •(I) E + K(a) + K(B) ~ 1 + K(a) + K(p) ( i i ) K(a) n K ( 0 ) ~ K(cT) n K ( p ) . ( i l l ) there exists r ^ g , s ^ , s 2 e R such that r j s 2 " r 2 s l ^ 0 a n d K^s^a-x^S) + p(s 2a-r 2 P ) ) > £ + K(a) + K(^) . We note that the development of these i n v a r i a n t s follows the general l i n e s of the developments described i n X \ section 2. A basis Is chosen and the i n v a r i a n t s are developed. \ The ef f e c t of a change i n basis i s determined and an equivalence i s defined on the system of Invariants. Here, as In section 29 the problem of determining the equivalence classes prevents the invariants from being a . r e a l l y s a t i s f a c t o r y c l a s s i f i c a t i o n . However the invaraints of Beaumont and Pierce are amenable to computation and thus do have several u s e f u l a p p l i c a t i o n s as we mentioned previously. One of these applic a t i o n s i s that the i n v a r i a n t s proyid necessary and s u f f i c i e n t conditions f o r a rank-two group to be d i r e c t l y decomposable and f o r a rank two group to be quasi- decomposable. Two s u f f i c i e n t conditions f o r a rank two group to be strongly indecomposable are also provided. C l e a r l y a. rank two group G is decomposable jntothe direct sun of two rank one-groups i f f G contains a basis xi>x2 s u c n t h a / f c (G^x^Xg) -• ( 2> x) where E a 0 . Hence a rank two group G i s quasi-decomposable i f f / • • G contains a basis x 1 » x 2 such that (Gjx x,x 2) (E,X) where E ~ 0 . This may be restated as: I f (Gjx-^Xg) -» (£,X), (a,0) e ithen G i s qua si-decomposable i f f there e x i s t r x * r 2 * s i ' s 2 e R with r^Sg.- r 2 s l ^ 0/.and K ^ a - r ^ p ) + K(s 2a-r 2B) > E + K(a) + '• / ' / • Application of the above statement r e s u l t s i n the two s u f f i c i e n t conditions f o r a rank two group to be strongly indecomposable. Let (G;x,,x 0) (E,X) , (a,0) e X . Then G • • / • ; / ' i s strongly indecomposable i f either f o r some P > -2(p) •= » 41. and ^^/a(p) /la i r r a t i o n a l or three d i s t i n c t primes, PX>P2>P;5> exist with Z(p x) - S(p'2) = 2(p^) = w and P ^ p i ^ / c ^ x ) > p^ p2^/a(p 2) , ^ p 3 ? / a ( p 5 ) a 1 1 d i s t i n c t . There r e s u l t s are from Beaumont and Pierce [4] . I t i s appropriate to mention here that Beaumont and Pierce's invariants are not the only ones that are amenable to computation. Mai'cev's invariants have been employed by Prochazha [pO] to f i n d conditions f o r rank two groups to be d i r e c t l y decomposable. We w i l l not go i n t o his r e s u l t s here but w i l l just say that he has discovered c r i t e r i a f o r d i r e c t decom- p o s a b i l i t y of a rank two group. He has a l s o found two sufficient; conditions f o r a rank two group to be indecomposable* which are analogous to the conditions f o r strong indecomposability given above. / The r i n g of quasi-endomorphistns 'E(G) of a group G i s a quasi-isomorphism inv a r i a n t and, hence i t i s desirable to determine E(G) . Beaumont and Pierce have used t h e i r i n v a r i a n t s to do t h i s f o r rank two groups. Let the type number of group G be the c a r d i n a l i t y of T(G) . Roughly t h e i r approach I s as follows: I f G i s a rank two group then the type number of G i s either one, two, or greater than two. In each case the invariants must s a t i s f y one of two or three mutually ex c l u s i v e , exhaustive conditions. ; E(G) i s then determined i n each case. The r e s u l t i s that E(G) must be (isomorphic to) one of R ; R + R (ri n g d i r e c t sum): a quadratic f i e l d over R ; the r i n g of a l l 2 x 2 - tr i a n g u l a r matrices i n R ; the r i n g of a l l 2 x 2 triangular matrices i n R with equal diagonal elements; the ri n g of a l l 2 x 2 matrices i n R . Examples are presented i n each case. Another method of determining E(G) f o r rank two • groups, due to Reid [j>k] 9- w i l l be presented i n section 8 . ReidS end r e s u l t s are exactly the same as those of Beaumont and Pierce. The question of when quasi-isomorphism implies isomor phism i s of obvious Importance. In the rank two case Beaumont and Pierce's invariants provide an answer. Use i s made of a r e s u l t by Baer [2] , namely; i f G £ G 1 (±) G 2 where G1 and G g are rank one groups of comparable types then any f i n i t e extension of G i s isomorphic to G ( i . e . any group quasi-isomorphic to G i s isomorphic to G ). A p p l i c a t i o n of t h i s r e s u l t and the above mentioned determination of E(G) leads tx>- the following r e s u l t . / : • ' " Theorem 5.7 Let (G;x 1,x 2) -* (£,X) / (a.,?)' e X . Then G has : the property that G s H implies G =• H i f f e i t h e r \ £(p) + hp(a(p)) + hp(3(p)) » f o r a l l primes p or G i s : \ quasi-isomorphic to a d i r e c t sum of two rank one groups of comparable types. \ ' . • . While we are speaking of conditions f o r quasi-isomorphic groups to be Isomorphic we should leave rank two groups f o r a moment and mention a conjecture of Beaumont and Pierce concerning groups of a r b i t r a r y f i n i t e rank which goes as follows: I f G i s a group of rank n containing a free subgroup F of rank n v 43. such that f o r each p", the d i v i s i b l e part of the p-primary component of G/F has rank at l e a s t n - 1 , and i f H £ G then H = G . That t h i s i s i n / f a c t true has been proved by / Prochazha [32] . His proof makes use of Mal'cev's Invariants. In another paper [29] , Prochazha has proved that we may replace the condition that f o r each p , the d i v i s i b l e part of. the p-primary component of /G/F has rank at l e a s t n - 1 by the condition that f o r each p , the p-rank- of G/F Is at l e a s t ' ; n-1 . j • • Discussion of the type sets of rank two groups w i l l be presented i n the next section along with some r e s u l t s on type sets of groups of a r b i t r a r y f i n i t e rank. Reids computation of E(G) for rank two' groups w i l l be described i n the. f i n a l s ection to demonstrate how the notions of irreducable groups and the psuedo-socle a f f e c t the structure of E(G) . / ; • 6. Type Sets / i h e type set of a group i s a quasi-isomorphism Invariant. As a r e s u l t I t would be h e l p f u l to have a necessary and s u f f i c i e n t . condition f o r a set of types to be the type set of a group of f i n i t e rank. However no such condition Is yet known. Some necessary conditions and some s u f f i c i e n t conditions are '.known. Beaumont and Pierce have used t h e i r i n v a r i a n t s to deterr>''; mine conditions f o r the rank two case. Dubois [ 1 0 ] , [ l l ] has applied a n a l y t i c number theory to the same case. The case of ar b i t r a r y f i n i t e rank has been examined by Koehler. [22] from a l a t t i c e t h e o r e t i c a l point of view. When v/e say that T(G) i s a quasi-isomorphism i n v a r i a n t we mean that i f G = H then T(G) ... T(H).. The converse, however, 44. i s not true i n general. We demonstrate t h i s by presenting a theorem and an example from Beaumont and P i e r c e . The theorem determines T(G) i n terms of the i n v a r i a n t s (£,X) . For B e " R ^ we define a c h a r a c t e r i s t i c A(B) by A(B)(p) - 0 , i f B(p) |= 0 and A ( B ) ( P ) - » i f B(p) - 0 . I f (Gjx ^ X g ) - (Z,X), (a,B) e X , we define a e ^ 1 ^ ^y a (p) « ^p}/a(p) i f hp(B(p)) > h p(a(p)) < « , a (p) « 0 otherwise, Theorem 6.1 Let (G;x 1,x 2) - (2,X) , (a,B) e X . Then T(G) = { [ i n ( K(o - s ) + A.(B - s a ) ) •+ ' ( K ( a ) n K ( B ) ) ] | seR, s^O} U {[K(a)], [K(f3)]) . . Example 6.2 Choose a fi x e d prime p and l e t ;ap be an I r r a t i o n a l element of R ^ with h ( a ) = 0 . Now define: a(q) - 1 f o r a l l q e TT J B(q) = 1 i f q f- p, B(p) = q p j E(qj = 0 i f q j. p, s(p) « « Then there exists a group G with,',-. ' a basis x 2 _ J X 2 S U C N ' T R I A ' F C ( G J x]_a x 2 ^ "* ( 2 * X ) ' (a>&) e X . Theorem 6.1 can be used to show that T(G) consists of the zero type alone. Furthermore i t i s possible to show that G Is strongly '.indecomposable. Let H be a rank two f r e e group. . Then "•'•},'; ,~ r T(H) m T(G) but G £ RV. Before proceeding further with the d i s c u s s i o n of type sets we define to notion of a quotient d i v i s i b l e (q*d.) group. Such groups are important i n the study of t o r s i o n free rings ( c f . Beaumont and Pierce [3]) . We define them here because 45. many of the results and examples Involve them. Definition 6.3 A'group G i s said to be a q.d. group i f G contains a f u l l /subgroup F, with F free, such that G/F i s the direct sum/of a divisible group and a group of bounded order. We note that i f G has f i n i t e rank then the group of bound/ed order i s f i n i t e . The following properties of q.d. groups are proved in Beaumont and Pierce [3] . If G i s q.d. and G k H then H i s q.d. Thus the property of being a q.d. group i s a quasi-isomorphic invariant. If G i s q.d. then G has a : free subgroup F such that G/F i s d i v i s i b l e . We w i l l also need the notion of a non-nil type. A characteristic i s said to be non-nil i f i t i s almost everywhere 0 or » . A type i s said to be non-nil i f i t contains a non-nil characteristic. I f t i s y a non-nil type then there exists a unique 9 e t such,that ©(p.)=0 or 00 for a l l p . Beaumont and Pierce [4] have used their invariants for rank two groups to characterize the rank two q.d.. groups. The result i s that i f (Gjx^Xg) - (£,X), (a,&) e X then G i s a q.d. group i f f Z + K(cc) + K(B) and K(a) H K(p) are non-nil: (as a consequence the group G of example 6 . 2 i s q.d.) In our discussion we w i l l also use the following d e f i n i - tions and notions. For any group. G , the "type number" of G i s the cardinality of T(G) and IT denotes the minimal" d i v i s i b l e group containing G . 46. Let C(G) - T(G) w {all f i n i t e i n t e r s e c t i o n s of members of T(G) J . Then we w i l l write P(G) «= {Gt | t e C(G) ) and Q(G) = (5 | t « C(G) ) We note that ( [ 1 3 3 ) G t Is a pure subgroup of G f o r a l l types t and that I f G has f i n i t e rank then P(G) i s countable. I f there i s no p o s s i b i l i t y of confusion we w i l l sometimes write G^ to denote . k A group G i s said to be completely " a n i s o t r o p i c i f no two Independent elements of G have the same type. The remaining d e f i n i t i o n s and notions are from Dubois [103 and [113 . Let C denote the set of a l l coprime ordered \ pairs of Integers . (a,b) with 0 _< a . Well order C by the ' \ re l a t i o n : I f max [a, | b| } _< max {c,|d|} then (a,b) precedes (c,d With t h i s well order we w i l l c a l l C the standard l i s t . Suppose that t Q Is a type, T a set of types, S : t ^ , t 2 , . . . a sequence of types and G i s a group. T i s , ti t it , II said to be a t Q - s e t i f f o r t , t e T, t <{= t , t n t = t . S i s said to be a t Q - sequence i f , f o r 0 < i < j , t ^ n t j « t S i s said to be a type sequence of G i f , f o r x and y independent i n G, C can be indexed so that t ( a x+b y 3G) <= t • * for a l l n . I f S Is a type sequence of a rank two group G then T(G) «= [t, | i ='1,2,.*.} . , Next, l e t S : t 1 , t 2 , be a t Q-sequence with 9 i 6 tl ( i = 0* 1* 2*' ••)' • w e write D(i,j,p) to represent the > proposition © Q(p) < ^ ( p ) < Qj(p)..- » . I f - i s a term of 47. S and S* i s a subsequence of S then 9^ i s said to be a i M i snarl of S i f there exists a subsequence S of S • , say ti S : 9 , 9 ..... such that for every k there i s a prime p v n l n 2 • with D(i,n k,p k) . 9 j i s said to be a snarl i f i t i s a snarl of some subsequence. A subsequence with no snarls i s said to be free. Finally, l e t T be a t -set with 9 Q e t Q A type [9] e T i s a snarl of the subset T* c T i f T* contains an II „ . i t i n f i n i t e subset T such that for every [9 ] i n T there » i i s a p e ir with 9 Q(p) < 9(p) < 9 (p) - » . A subset T of T i s said to be free i f i t has no snarls. • The last of our preperatory notions i s the construc tion, due to Dubois [11] , of groups R(S,x) which are useful- in type considerations. S i s an independent set of reals i n the open interval (0,1) and x i s a function-from S Into TT 2 ; ( p ) such that, for s e S , x(s) i s a function on ir whose value at p i s a p-adic Integer. We write. x(s) (p) = x(s,p) v-.^ We define R(S,x) to be the set of a l l f i n i t e rational combina tions £ r s (r eR) such that for every p, 2 r x(s,p) e \ Then R(S,x) i s a group vjith rank | S| such that lip(2 r g s , R(S,x)) mt hp(E r gx(s,p)) and for every p ; the cor respondence 2 r s -• 2 r x(s,p) i s a p-height preserving S S i . . . I homomorphism with kernel equal to the set of a l l members of R(S,x) with i n f i n i t e p-height and a p-pure image i n Z^p^ . 48. In the case where S = [x,y] , we denote the func tions by u and v and the group hy R(x,y;u,v) . Then R(x,y;u,v) i s the group of a l l r a t i o n a l combinations ax + by where for a l l p au(p) + bv(p) e Z^p^ . We now present necessary conditions on a type set to be the type set of a group G . Beaumont and Pierce [ 4 ] have used th e i r invariants to examine the rank two case. Let G be a rank two group and suppose that x x,Xg i s a basis of G and ( G J X X , X 2 ) - ( 2,X), (a,p) e X . Then t(x x,G) fl t(x 2,G) - ' • [K(a) n K(0)] . Theorem 6 . 4 I f G i s a rank two group there i s a unique type tQ'; such that i f t - ^ t g e T(G) , t± ± tg , then t x n tg « t Q . I f x and y are non-zero elements of G with t(x) = t(y) t - then x and y are dependant. I f T(G) I s f i n i t e then 11 e T(G) I f r(G) « 2 and G has f i n i t e type number then G i s not completely an i s o t r o p i c . Thus by theorem 6.4 i f t Q £ T(G) then G i s completely a n i s o t r o p i c . Note that theorem 6 . 4 can be restated as: i f r(G) = 2 then T(G) i s a t Q - s e t . Dubois X l O ] has also obtained t h i s r e s u l t along with: every type sequence of G i s a t 0-sequence. ( A c t u a l l y he has provided a stronger necessary condition. Employing some ideas from a n a l y t i c number theory he proves | I Theorem 6 . 5 I f G i s a rank two group then the type sequence v of G has an i n f i n i t e free subsequence .• I f G i s completely 4 9 . anisotropic then T(G) contains an i n f i n i t e free subset. An example vd.ll demonstrate that the necessary condi t i o n of t h i s theorem i s stronger than that of the preceding one. The same example also provides a strong negative answer to a question posed by Beaumont and Pierce, namely: Given a count- ably i n f i n i t e set T of d i s t i n c t types such that T i s a t Q - s e t does there exist a rank two group G with T(G) = T ? Example 6.6 [10]' Suppose the i n f i n i t e sets P 1,P 2,... p a r t i t i o n TT . Ue define ©-^p) = 1 i f p e P x , 6 1(p) <= 0 elsewhere and, • *f* for n « 2,3,... , e n ( p ) = » i f p is, the n member of P i f o r .some. . i < n , e n ( p ) 1 i f P € p n> * 0 otherwise. Then T = £ C ] , [© 2],...} i s . a t Q - s e t where t Q , i s the zero type and, by theorem 6.5 i f T* i s any subset of T , then T* i s " " not the type set of any completely an i s o t r o p i c rank two" group. Dubois has also applied the notion of groups of the form R(x,y;u,v), to f i n d a necessary and s u f f i c i e n t condition f o r a t Q-sequence to be the type sequence of a group. In [ l l ] an example i s constructed to prove that every type sequence of a rank two group; i s a type sequence of some rank two group R(x,y;u,v) . Examination of type sequences of a r b i t r a r y groups R(x,y;u,v) w i l l y i e l d ; 1 . ; • Theorem 6.7 A t^-sequence i s a type sequence of a rank two group i f f the zero sequence obtained by subtracting -t • from every term i s likewise, a. type sequence of a rank two group. ' Koehler [22] has obtained necessary conditions on type sets of groups of f i n i t e rank. His approach i s to show that the type set of a f i n i t e rank group has c e r t a i n l a t t i c e s of types and of pure subgroups associated with i t . His f i r s t r e s u l t i s to prove that, i f r(G) = n , then C(G) forms a l a t t i c e of length at most n i n which l a t t i c e meet i s type i n t e r s e c t i o n and C(G) has a minimum type t where t = t ^ j n ... D t ( x n ) for any basis x 1,x 2,...,x n of G . The set of a l l types under the r e l a t i o n £ and the operations fl and U forms a d i s t r i b u t i v e l a t t i c e i n which meet and j o i n are fl and U respectively. I f G has f i n i t e rank then the above remark t e l l s us that C(G) i s a l s o a l a t t i c e . However i t need not be a s u b l a t t i c e of the l a t t i c e of a l l types (example 6.15) . Note that t h i s remark i s a g e n e r a l i z a t i o n of theorenT"" 6.4 . This remark also provides an aff i r m a t i v e answer to another question posed by Beaumont and Pierce [4] : I f r(G) = n is the i n t e r s e c t i o n of the types of elements of a given basis the same for a l l bases? Furthermore, suppose G Is a f i n i t e rank q.d. group and that -. t i s the minimum type i n C(G) . Then ( [ l l ] or [22]) 't i s non-nil. In t h i s connection we also have ( [ l l ] ) that the type set of rank two group G i s the type set of some q.d. group i f f t Q , the minimal type i . i . . . i n C(G) , i s non-nil. I i • I _ _ Theorem 6.8 I f G i s a group of f i n i t e rank n , jthen G = R and j . ( i ) P(G) forms a l a t t i c e of pure subgroups.of G , •'• Q(G) forms a l a t t i c e of subspaces of R n and as l a t t i c e s , P(G) i s isomorphic to Q(G) ' and both are 51. dually isomorphic to C(G) . ( i i ) In the l a t t i c e s P(G) and Q(G) I f A and v denote l a t t i c e meet and j o i n respectively then, f o r G ±,Gj € P(G) G ± A Gj - G ± n G j ; \ A JS^ = "G± n' tf j This theorem, due to Koehler , can be used to prove that i f T(G) i s f i n i t e then T(G) •= C(G) and there are r(G t) independent, elements of type t i n G f o r every t e T(G).. For an example of a group of f i n i t e rank and i n f i n i t e type set T(G) such that T(G) J* C(G) see Beaumont and Pierce [4, pg. 29] . This concludes our review of necessary conditions on a type set to be the type set of a group of f i n i t e rank. We now turn to the problem of f i n d i n g s u f f i c i e n t conditions. As; with the necessary conditions we f i r s t turn our a t t e n t i o n to the rank two case. In the case of f i n i t e type sets we have a complete answer. ; Theorem 6.9 Let T = i^Q^'i? • • • b e a f i n i t e t Q - s e t . Then thei r exists a rank two group G with T(G) = T \ Both Beaumont and Pierce [ 4 ] and Dubois , [10] have^ constructed examples to prove t h i s r e s u l t . As a r e s u l t of \ theorem 6.4 and 6.9 we have ! ' \ • V ; . * . ,-. . ; ; , ^ Corollary 6.10 A f i n i t e type set T i s the type set of rank two group i f f T i s a t Q ~ s e t containing t Q , . I 52. Suppose T i s a f i n i t e t 0 - s e t containing t Q and t Q i s non-nil. Then does there e x i s t s a q.d. rank two group G with T(G) = T ? This question was.posed by Beaumont and Pierce [ 4 ] . I f (Gjx - ^ X g ) - ( E , X ) , (a,p) € X then G i s q.d.; i f f Z + K(0) -f K(a) and K(a) n K(0) are non-nil. In the question as posed t Q = [K(a) 0 K(0) ] i s given as non-nil and so to answer the question, a group G must be found, with T(G) = T such that G contains a basis x-]_,x2 with (G;x x,x 2) -» (Z,X) where £ i s non-nil and £(p) = « f o r a l l but a f i n i t e number of primes p such that 0 < hp(a(p)) < « and 0 < h p(p(p)) < whenever (a,0) e X . That such a group does e x i s t has been proved by Koehler [21] . This question has also been answered by Dubois [10] who has construced an example to prove that: A free t Q-sequence, with t Q non-nil, i s the type sequence of a q.d. rank two group. ' In the case of countably I n f i n i t e type sets Beaumont and Pierce were only able to achieve p a r t i a l r e s u l t s . They have constructed an example of a completely a n i s o t r o p i c rank two group G with type set T(G) £ [ [ G o ] , [ ^ 1 ] , [ G 2 ] , . . . } where {^ Q,\>^ 2 9 *' ' i s an i n f i n i t e set of inequivalent c h a r a c t e r i s t i c s with, f o r i L £ e. 0 9, • e . and used i t to prove: I f — T i s an i n f i n i t e t Q - s e t containing t Q and each type i n T i s f i n i t e then there i s a completely anisotropic rank two group G such that X 5 3 . Koehler [21] has developed a new method of con structing rank two groups with i n f i n i t e type sets which enables him to prove some p a r t i a l r e s u l t s s i m i l a r to those above, namely: I f T - [ t Q , t 1 , t 2 , . . . } i s a t Q - s e t then there e x i s t s a t Q - s e t T m { t o , t ^ , t 2 , . . . } ' with t i < t'± i f i >. 1 and a rank two \ group G with T ' C T(G) . I f T = [ t Q , t - ^ t g , . . . } i s a t 0 - s e i \ such that f o r at most f i n i t e l y many i © i(p) =» <» occurs where 6 i e t^ then there ex i s t s a rank two group G with' T c T(G) . In some cases Koehler's construction y i e l d s good r e s u l t s . For instance, i f we define © 0(p) = © f o r a l l p ; ^ ( p ) - 1 f o r a l l p ; e k ^ p k - l ^ " 0 9' 9k^ p^ = ° f o r a 1 1 0 - t n e r P , then T = {[O ],[© 1],[Gg],... ) i s a [9Q] set and the rank two group G r e s u l t i n g from the construction has type set T(G) « T . Dubois has obtained some precise r e s u l t s i n determining . . . / • • • ( . . s u f f i c i e n t conditions f o r a type set to be the type set of a rank two group. In [10] a n " example i s given which proves; ' Theorem 6.11 / (i) I f S i s a free t Q-sequence then S i s a type sequence of some rank two group.j ( i i ) I f T i s an i n f i n i t e -'t -set whose'members are i n f i n i t e only where t Q i s i n f i n i t e then T . i s the type set of a rank two group. We now present an example due to Dubois which demons-s / • trates that the s u f f i c i e n t condition of theorem 6.11 Is not 54 necessary. This example w i l l a l so provide a negative answer to another question posed by Beaumont and Pierce [4] : I f G i s a group with i n f i n i t e type set does there e x i s t s a completely anisotropic group H with T(H) « T(G) ? We define f o r n = 0,1,2,... : e 2 n ( p ) = 0 f o r a l 3 L p 5 9 l ( p ) = 1 f o r a 1 1 p > for n > I , ©2n + 1(p) =0 f o r p ' p n , 0 2 n + 1 ( p n ) - - . • Then S : [ © - [ _ ] , [ © g ^ * * * i s a [0 o]-sequence and, i f T'= {[©iJ | i = 1,2,...} , [G^] i s a snarl of every I n f i n i t e subset of T . Hence T i s not the type set of a completely anisotropic rank two group. Also neither S nor T i s f r e e . ,; Now l e t c x = (1,0), c2«=(0,l) . For n > 1 l e t Cg^J. = (^-P^) and c2n-t-2 e 1 u a l to the f i r s t p a i r i n the standard l i s t n o t p r e - viously selected. We write c n » (a n,b n) . Choose independent^ re a l numbers x a n d y and l e t G « {ax+by |. a,beR, ap .+ b e Z ^ ' ^ • f o r a l l p} . Then G i s a rank two group. Note that i f x,y e (0,1) then G i s a group of the form R(x,y;u,v) where u(p) = p and v(p) = 1 f o r a l l p . We also have, f o r a l l k « 1,2,... [e^] = t(a kx+b 1 (y ,G) . Hence S i s the type sequence of G and •T = T(G) . In [11] , Dubois strengthens the s u f f i c i e n t c o n d ition of theorem 6.11. A. c h a r a c t e r i s t i c i s said to be very large i f i t i s i n f i n i t e at i n f i n i t e l y many primes. / I Theorem 6.12 I f a t -sequence, r S;, has an i n f i n i t e f r e e 55. subsequence and i f the set of a l l snarls and very large elements of S i s free then S i s a type sequence of a rank two group. The group constructed to prove theorem 6.12 i s a group of the form R(x,yju,v) . Dubois has constructed another group R(x,y;u,v) which demonstrates that the condition that the set of a l l very large elements be free i s not necessary. The various s u f f i c i e n c y theorems of Dubois, e s p e c i a l l y theorem 6 . 7 and the remark immediately preceeding i t , suggest the following formulation of the problem of determining necessary and s u f f i c i e n t conditions i n the rank two case. Let 0 : © ^ © ^ J * be a sequence of c h a r a c t e r i s t i c s with the corresponding sequence of types, S , a zero-sequence. Such a sequence 9 i s said to be solvable i f there e x i s t s an indexing of. the elements (a,b) of C so that f o r every j I there e x i s t s an.- m such t h a t / f o r a l l — p G TT and . n > m ,j- i f e n ( p ) • « then ^p'C^^n"8-^!')' ^ ^ . ( P ) • Then, with S and © as above . ; Theorem 6 . 1 3 S i/s a type sequence of a rank two group i f f 0 i s a solvable sequence. Koehler has found a s u f f i c i e n t condition f o r a f i n i t e \ / . • ' • • • •type set to be /the type set of a f i n i t e rank group. Let t be • / ••• the type greater than a l l types. ( Theorem 6.14 Let T = ft^t jt-^j .. .,t N} D e a set of d i s t i n c t i types forming a l a t t i c e under A and v where t. /\ t . «=t. f i t . i • J i J and t ± v t j i s the l.u.b. i n T . Let ;L = {0,^,^,.. ,p N] be 5b. a l a t t i c e of subspaces of R n ="&"_, under A and v where ')• / G. A IT = G~. fl Gj and v i s the l.u.b. i n L . Suppose that, as 1 j 1 P - l a t t i c e s , T i s d u a l l y isomorphic to L . Then there>exlsts a rank n group G with T(G) = T and Q(G) =.L . The r e s u l t s of theorems 6 .8 and 6.14 Indicate that the problem of f i n d i n g a l l f i n i t e type sets which are type sets of a group of f i n i t e rank n. i s equivalent to the problem of fi n d i n g a l l possible f i n i t e l a t t i c e s , under the operations A : and v, of subspaces of R n * This l a t t e r problem i s as yet unsolved. '*.'., \ Example 6.15 We define 9 Q(p) = 0 for a l l p 5 e x ( 2 ) • > © x(p) = 0 otherwise; © 2 ^ ) = 00> Q 2 ^ " 0 °"tnervriLse^ 6_(2) =-6_(3) 6 (5) = » , 9,(p) <= 0 otherwise . I f we set_„ p P P , P . / . . t^ = [9^3 then by theorem 6 .14 , since the dual of the l a t t i c e of types i s r e a l i z a b l e i n R^ , there e x i s t s a rank 3 group G with T(G) « C(G) - {t<a»t0>ti>t2't^ ' H e n c e C ( G ) forms a l a t t i c e i n which v t g = > ^ U t g , Hence C(G) i s not a su b l a t t i c e of the l a t t i c e of types. \ We conclude our discussion of type sets with the following r e s u l t from Dubois [113 . ' 1. • ' ' ' • ' • ! Theorem 6.16 I f T i s a set of types with the property that t , t e.T implies t (IT e T then T i s the type set of a group R(S,x) of rank |T|* . . • j N 57. • . 7. Quasi Essential Groups We devote t h i s section to a b r i e f discussion on the notion, due to Koehler [22] , of quasi- essential groups. As we/will see these groups can be used to suggest a possible approach to the problem of f i n d i n g quasi- isomorphic invariants' 7for groups of f i n i t e rank with f i n i t e type sets. The d e f i n i t i o n follows the construction of the group of theorem 6.14 and hence provides some idea of the method employed I there. / ' • / D e f i n i t i o n 7*1 Let G be a group and l e t B = {x^,...xn} be any f i n i t e set of independent elements of G . Let Fg denote the free subgroup of G generated by - B . An element x e G i s said to be /B-reduced i f x e Pg and hp(x,F B) = 0 f o r a l l P e TT . • J - D e f i n i t i o n 7.2 A group G of f i n i t e rank i s said to' be an essen t i a l group i f i t has for. a set of generators, the set . -s (p) k . {P y± I pelf , 0 < s k(p) < © k(p)+l; k-0,l,...,N 5 i=l,2,...,n k] where (i) . © o , 0 1 , . . . , 0 N are c h a r a c t e r i s t i c s with «= [9^] such that i f t± < t j - then 9± _< ©^ and i f t± fl t j ' • t f c then © "0 .©., = ft , . © < i , j , k < N ( i i ) r i k = r ( G t )(=r(G^) ; k = 0,1,... ,N . . i k ( i i i ) B Q = {y£,y°,...,y° } i s a basis of G such that y° £ "G"k i , 1 1 k i N » 1 < i < n 0 • 1 (iv) f o r each k » 1,2, ...,N , {y^*y|* *.-..*y^ } i s a ba s i s 58. .of G K such that y k ' i s B Q-reduced and y£ £ If" i f T?j c G K . . A group H i s said to be a quasi-essential (q.e) group i f i t i s quasi-isomorphic to some e s s e n t i a l group G . Suppose that y. e R n and that 0 are c h a r a c t e r i s t i c s A. X (\cA) . By the notation G = {(y^,©^) I XeA} we w i l l mean the group G generated by the set {p" SX^ p^y^ | peir ; 0 _< s^(p) <, 9 (p) + 1 ; \eS) . Hence i f G i s the group of d e f i n i t i o n . 7.2 then G = {(y£,© k) }. Furthermore T ( G ) { t ^ t ^ t ^ ... , t N ) and Q ( G ) = [ 0 9 \ , \ 9 . , \ ) . D e f i n i t i o n 7 .3 Let G * be an es s e n t i a l (q.e.) subgroup of a group G . G ' i s said to be a maximal e s s e n t i a l (q.e) sub- f • * ' * * group i f G c H £ G where H i s an e s s e n t i a l (q.e)/ subgroup- of G then G 4 H . . . • ' "'. The p r i n c i p a l r e s u l t on these groups, f o r our purposes, i s the following: Theorem J.H- Let G be a f i n i t e rank group with f i n i t e type set. Then l(i) G has a maximal e s s e n t i a l subgroup G * , unique up to quasi-equality, with T ( G ) = T ( G * ) and Q ( G ) * Q ( G ' ) . ( i i ) I f x e G there Is a maximal e s s e n t i a l subgroup G * of G with x e G . ( i i i ) G i s q.e. i f f G / G * i s a f i n i t e group f o r |every maximal essential subgroup G of G . (iv) I f G i s a maximal e s s e n t i a l subgroup of G then f . . G / G i s a t o r s i o n group. ';. • 59. P r o o f : We w i l l p r o v e ( i ) and ( i i ) . L e t r ( G ) = n and T ( G ) = { t e o , t o , t x , . . • w i t h t Q t h e m i n i m a l t y p e i n T ( G ) . There i s an indepe n d e n t s e t [x,y°,... ,y° 3 w i t h t(y°) » t Q . I f t ( x ) = t Q l e t y° = x . O t h e r w i s e l e t S be t h e p u r e sub group o f G ge n e r a t e d by (x>y23 . Now t ( y , S ) = t ( y , G ) f o r a l l y € S and t(y°,S) = t . F o r some m € Z t t x + m y ^ S ) = t Q « t(x+my°,G) . L e t y° - x + my° and B q . [y£,y°,•'... ,y°) . Then x i s B -re d u c e d . o Fo r each t R € T(G) , t f c f= t . we can f i n d n R - r ( G k ) k k k independent B o - r e d u c e d elements o f t y p e t k I n G, y 1 , y 2 > • • • * y n . v -1c D e f i n e 9 f c « J h(y^,G) f o r k •= 0,1, ...,N . I t i s p o s s i b l e t o f i n d c h a r a c t e r i s t i c s 9^,6^, . . i ,9^ s u c h t h a t , f o r 0r_< i , j , j jC-N i < * i i % - \ •; i f t± < t-•• t h e n •;. 9' ;< ;e* • j i f t ± n t j = t k I 9 the n 9^ 0 9^ = 9 k . L e t G * m C ^ , 9 k ) | k « 0,1,...,N;; X= l , 2 , . . . , n k 3 . G * I s an e s s e n t i a l subgroup o f G and T ( G ' ) '« T ( G ) T ^ Q ( G ' ) » Q ( G ) . Furthermore G i s maximal e s s e n t i a l , c o n t a i n s x and i s \ unique up t o q u a s i - e q u a l i t y . The r e s u l t s o f t h i s theorem s u g g e s t t h a t t h e p r o b l e m we mentioned i n t h e o p e n i n g p a r a g r a p h o f t h i s s e c t i o n c o u l d p o s s i - ' b l y be s o l v e d by examining, t h e groups o f t h e f o r m G/G • where G has f i n i t e r a n k , G i s a maximal e s s e n t i a l subgroup o f G . 61. (i) G i s irreducible. ( i i ) G = ^ © G . where each G, i s strongly indecomposable, i=l • irreducible, and G i k G^ for a l l i , j . ( i i i ) E(G) = A n where A i s a division algebra, n i s the number of strongly indecomposable summands i n a quasi- decomposition of G and n[A:R] m r(G) . An irreducible group of f i n i t e rank i s strongly indecomposable i f f E(G) i s a division ring. An irreducible group of prime rank i s either strongly indecomposable or a direct sum of isomorphic rank one groups. If E(G) has descending chain condition on right ideals then P i s non-zero. Also i f G i s strongly indecomposable and N denotes the. radical of E(G) then as groups /\;:-". r(H) = r(H) « [E(G)/N:R] , where H Is a 'minimal non-zero pure fu l l y invariant subgroup of G . If E(G) has descending chain' condition on right ideals then E(G) i s semi-simple i f f G « P . If G has f i n i t e rank and G «j= P then for x e G , x £ S there exists an endomorphism X of G with 0 ^ x\ e S . Now suppose that G i s a rank two group. ; Then G must satisfy one of three mutually exclusive exhaustive conditions namely;. G i s irreducible, G i s not Irreducible but G = P^ or G f P. Suppose G i s irreducible* If G Is strongly indecom-62. posable then E(G) i s a division ring A . Furthermore 2 [A:R] a.2 . If B i s the center of A then [A:B] = x where x i s a positive integer. ; Now [A:R] = [A:B][B:R] and so [A:B] a 1 . Hence A i s equal to Its center and i s thus a quadratic f i e l d over R . If G i s not strongly indecomposable then G a G x0G 2 where Q1 and G 2 are isomorphic rank one groups. In this case E(G) a Ag where A i s a division algebra and E(G1) = A (i=l ,2) . Since G± has rank one, A » R , and E(G) = R 2 , the ring of a l l 2 x 2 matrices i n R . Next suppose that G i s not irreducible and G = P . If G i s strongly indecomposable, then E(G). i s a division algebra. Also G contains a minimal non zero pure f u l l y i n v a r i ant subgroup of rank one. Hence [E(G):R] a 1, i . e . E(G) ,= R . If G i s quasi-decomposable then E(G) i s a semi-simple ring — that i s not a division ring. If E(G) were simple then G would be irreducible. Hence E(G) = E.^ + Eg (ring direct sum) where the E^ are simple ideals with central Idempotent gener ators. Corresponding to this decomposition E(G) a E^ +. Eg we have a quasi-decomposition G « G^ @ Gg where the are 'rank one groups of incomparable types. Also E(Gx) £ E^ and therefore E ± = R (i=l ,2) . Thus E(G)" - R + R . Finally suppose that G ^ P . Assume. G i s strongly indecomposable. Choose 0 =f= x^ e P , x g e G , x g ^ P . Then { X i j X g } i s a basis, of, V , which i s an E(G)-module i n a natural ' 6 2 . way.' P i s an E ( G )-submodule of V and so the m a t r i x r e p r e se n t a t i o n of E ( G ) given by . ( x ^ X g ) c o n s i s t s - o f t r i a n g u l a r matrices. N > the r a d i c a l of E ( G ) , i s non-zero and I s a r a t i o n a l a l g e b r a . Hence N = [(° | . reR} . We a l s o have t h a t E ( G ) / N a R . Let P denote the subalgebra of E ( G ) g e n e r a t e d ^ by the u n i t m a t r i x . Then N n F = 0 . Under the n a t u r a l map F goes onto E ( G ) / N and so E ( G ) ~ N + F . Hence E ( G ) i s the r i n g of t r i a n g u l a r matrices i n R w i t h equal d i a g o n a l elements* I f G i s quasi-decomposable then G = P 0 p ' f o r some P* . Furthermore the type of P* i s l e s s than the type of P . As before a t r i a n g u l a r r e p r e s e n t a t i o n of E ( ; G ) can be o b t a i n e d . The r a d i c a l N i s one-dimensional over R . Hence E ( G ) / N i s a two-dimensional semi-simple, r a t i o n a l a l g e b r a t h a t i s not a d i v i s i o n a l g e b r a . T h i s I m p l i e s E ( G ) / N = R 4- R and hence E ( G ) i s the r i n g of a l l , 2 x 2 t r i a n g u l a r m atrices I n R / ••' We note I n c o n c l u s i o n t h a t these are the same end r e s u l t s at which Beaumont and P i e r c e 7 [k] a r r i v e d i n t h e i r computation of E ( G ) f o r rank two groups* 64 9 . Bibliography 1 / [I] Armstrong, J.W.; On the indecomposability of t o r s i o n f r e e abelian groups. Proc. Amer. Math. Soc. 1 6 ( 1 9 6 5 ) , 223 -325. [2] Baer, R.; Abelian groups .without elements of f i n i t e order. Duke Math. J . 3 ( 1 9 3 7 ) , 6 8 - 1 2 2 . [3] Beaumont, R.A.; Pierce, /R.S.; Torsion free r i n g s . I l l i n o i s J . Math. 5(1961), 6 1 - 9 8 . [4] Beaumont, R.A.; Pierce, R.S.; Torsion free groups of rank two. Mem. Amer. Math. Soc. No. 38 ( 1 9 6 1 ) , 41 pp. / • * - / * / [5] Campbell, M.O'N.;/ Countable t o r s i o n free abelian group's. Proc. Lon. Math/ Soc. (3) 10(1966). , 1 - 2 3 . s [6] Corner, A.L.S./ A note on rank and d i r e c t decomposition of torsion free abelian groups. Proc. Cambr. P h i l . Soc. 57(1961) , 2 3 0 - 2 3 3 . [7] Corner, A.L.S.; Every countable reduced t o r s i o n . f r e e ring., i s an endomo.rphism r i n g . Proc. Lond. Math. Soc. (3) 13(1963) , 6 8 7 - 7 1 0 . . ,• • ' ] ' . [8] Corner, A.L.S.j On a conjecture of Pierce concerning d i r e c t decompositions of abelian groups. Proc. Colloq ;. Abelian groups (Elhany, 1963) pp. .43-48. .Akad emiai Kiado, Budapest, 1964. . / • . . - ' L9J Derry,./D.; Uber eine Klasse von abelschen Gruppen. Proc. Lon. Math. Soc. 43 (1937) , 490-506* ' [10] Dubois, Donald; Applications of a n a l y t i c number theory to the study of type sets of tor s i o n . f r e e abelian groups. I . Univ. New Mexico, Dept.; of Math. Technical Report No. 56(1964). [II] Dubois, Donald; Applications of a n a l y t i c number theory to the study of type sets of t o r s i o n f r e e abelian 'groups. I I . Univ. New Mexico, Dept. of Math. Technical Report No. 58(1964) . 6 5 . [ 1 2 ] Erdos, Jeno;/ Torsion free f a c t o r groups of fr e e abelian groups and a c l a s s i f i c a t i o n of t o r s i o n free abelian groups. Publ. Math./ Debrecen 5 ( 1 9 5 7 ) , 172-184.,; [ 1 5 ] Fuchs, L . / Abelian Groups. Publishing House of the Hungarian Academy of Sciences, Budapest ( 1 9 5 8 ) • [14] Fuchs,/ I i . ; The existence of indecomposable abelian groups of a r b i t r a r y power. Acta. Math. Akad. S c i . Hung. 1 0 ( 1 9 5 9 ) , 4 5 3 - 4 5 7 . • ' / ' [15] Fuchs, L.; Recent r e s u l t s and problems on abelian groups. (Proc. Sympos. New Mexico State Univ.; I962) pp. 9-40. Scott, Foresman and CO./ Chicago 1 1 1 . I 9 6 3 . [ 1 6 ] de Groot, J . ; de Dries, H.; Indecomposable abelian groups with many automorphisms. Nieuw. Arck. Wisk. ( 3 ) 6 ( 1 9 5 8 ) , 5 5 - 5 7 . • " • [17] Harrison, D.K.; I n f i n i t e abelian groups and homological methods. Ann. of Math. (2) 6 9 ( 1 9 5 9 ) , 366-391.' ' • ' / • , ' ' ' , • ' ' [ 1 8 ] Jesmanowicz, L'.'j On d i r e c t decompositions of t o r s i o n f r e e • abelian groups. B u l l . Akad. Polon. Sc'i. Ser. S c i . Math. Astronom. Phys. 8 ( 1 9 6 0 ) , 5O5-51O. [ 1 9 ] Jonsson, B j a r n i ; On d i r e c t decompositions of t o r s i o n f r e e abelian groups. Math. Scand. 5 ( 1 9 5 7 ) , 2 3 0 - 2 3 5 . [ 2 0 ] Jonsson, B j a r n i ; On d i r e c t decomposition of t o r s i o n f r e e abelian groups. Math. Scand. 7 ( 1 9 5 9 ) , 3 6 1 - 3 7 1 . [ 2 1 ] Koehler, John E.; Some to r s i o n free rank two groups. Acta. S c i . Math. (Szeged) 2 5 ( 1 9 6 4 ) , I 8 6 - I 9 O . • . [ 2 2 ] Koehler, John E.; The type set of a to r s i o n free group of f i n i t e rank. I l l i n o i s J . Math. 9 ( 1 9 6 5 ) , 6 6 - 8 6 .j [ 2 3 ] Kovacs, L.G.; On a paper of Lad i s l a v : Prochazka. Czechos lovak Math. J . 13 (88) (I963), 6 1 2 - 6 1 8 . : ' . . : \ 66. [24] Kurosh, A.G.; Primitive t o r s i o n f r e l e abelsche Gruppen vom endlichen Range. Ann. of Math. 3 8 ( 1 9 3 7 ) , 1 7 5 - 2 0 3 . [25] Kurosh, A.G.; The Theory of Groups I. Chelsea Publishing Co. New York, 1955. / [26] Levi, F.j Abelsche Gruppen mit abzahlbaren Element^n-. Dissertation, L e i p z i g , 1917. / [27] Mal'cev, A.I.; Torsion free abelian groups of f i n i t e rank. Mat. Sbornik, 4 ( 1 9 3 8 ) , 4 5 - 6 8 . [28] Pontryogin, L.S.; The theory of t o p o l o g i c a l commutative groups. Ann. of Math. 3 5 ( 1 9 3 * 0 , 3 6 I - 3 8 8 . [29] Prochazka, Ladislav; A note on quasi-isomorphism of torsion-- free abelian groups of f i n i t e rank. Comment. Math./Ajniv. Carolinae 3 ( 1 9 6 2 ) , no. 1, 1 8 - 1 9 . [30] Prochazha, Ladislav; Conditions f o r decomposition Into a di r e c t sum f o r to r s i o n free abelian groups of rank two. Mat. Fyz. Casopis Sloven. Akad. Vied. 1 2 ( 1 9 6 2 ) , 1 6 6 - 2 0 2 . r ~ [31] Prochazka, Ladislav; A generalization, of a theorem of 1 R. Baer. Comment. Math. Univ. Carolinae 4 ( 1 9 6 3 ) , IO5-IO8. [32] Prochazka, Ladislav; A remark on quasi^isomorphism of ; torsion free groups of f i n i t e rank. Czechoslovak Math. J . 15(90) ( 1 9 6 5 ) , 1 - 8 . [33] Reid, J.D.; On quasi-decomposition of t o r s i o n free a b e l i a n \ groups. Proc. Amer. Math. Soc. 13(1962), 55O-554. [34] Reid, J.D.; On the r i n g of quasi-endomorphisms. of a t o r s i o n free group. (Proc. Sympos. New Mexico State Univ. 1962) pp. 5 1 - 6 8 . Scott, Foresman and Co. Chicago, 111; (1963) . • ' • ' ! - " [35] Rotman, Joseph; Torsion free and mixed abelian 1 groups. I l l i n o i s J. Math. 5 ( l 9 6 l ) , 131-143. '! N' [36] Szekeres, G.; Countable abelian groups without t o r s i o n . Duke Math. J . 15(1948) , 2 9 3 - 3 0 6 . Of. [37] Walker, E.A. j- Quotient categories and qua si-isomorphisms of abelian groups. Proc. Colloq. Abelian groups (Tihany, 1963) . Akademlai Kiado, Budapest, 1964, , pp. 147-162 [38] Wang, John S.P.j On completely decomposable groups. Proc. Amer. Math. Soc. 15 (1964) , 1 8 4 - 1 8 6 .
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A survey of recent results on torsion free abelian groups Duke, Stanley Howard 1967
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Title | A survey of recent results on torsion free abelian groups |
Creator |
Duke, Stanley Howard |
Publisher | University of British Columbia |
Date | 1967 |
Date Issued | 2011-08-09 |
Description | This thesis is a survey of some recent results concerning torsion free abelian groups, hereafter referred to as groups. The emphasis is on countable groups, particularly groups of finite rank. Section 1 contains the introduction and some notation used throughout this thesis. We begin in section 2 by describing the general nature of the existing characterizations for countable groups and by describing why these characterizations do not provide satisfactory systems of invariants. We include here a brief description of a classification for groups of arbitrary power. Pathologies of groups are discussed in section 3. We briefly discuss rank one groups and completely decomposable groups and then present examples to show the vast number of indecomposable groups which exist and that a group may have two different decompositions into the direct sum of indecomposable groups. Quasi-isomorphism and the ring of quasi-endomorphisms of a group are introduced in section 4 and discussed briefly. We present the theorems which establish the importance of these notions; namely that (i) quasi-decompositions of certain groups are unique up to quasi-isomorphism and (ii) the quasi-decomposition theory of certain groups is equivalent to the decomposition theory of the quasi-endomorphism ring considered as a right module over itself. Included under 'certain groups’ are the groups of finite rank. Section 5 is devoted to rank two groups. We outline the development of the quasi-isomorphism invariants for rank two groups, due to Beaumont and Pierce, and discuss some of their applications. For example, conditions, in terms of the invariants, are given for quasi-isomorphic rank two groups to be isomorphic. Type sets are reviewed in section 6. We present both necessary and sufficient conditions for sets of types to be the type sets of rank two groups and of groups of arbitrary finite rank. We devote section 7 to a brief discussion of the notion and importance of quasi-essential groups. The ideas of irreducibility and the psuedo-socle are defined in section 8. We demonstrate how these ideas affect the structure of the quasi-endomorphism ring by showing how they can be used to compete the quasi-endomorphism ring of rank two groups. |
Subject |
Abelian Groups. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-08-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080553 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/36614 |
Aggregated Source Repository | DSpace |
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