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The tensor product of two abelian groups 1966
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Title | The tensor product of two abelian groups |
Creator |
Mitton, David |
Publisher | University of British Columbia |
Date Created | 2011-08-31 |
Date Issued | 2011-08-31 |
Date | 1966 |
Description | The concept of a free group is discussed first in Chapter 1 and in Chapter 2 the tensor product of two groups for which we write A⊗B is defined by "factoring out" an appropriate subgroup of the free group on the Cartesian product of the two groups. The existence of a unique homomorphism h : A⊗B→H is assured by the existence of a bilinear map f : A×B→H , where H is any group (Lemma 2-2) and this property of the tensor product is used extensively throughout the thesis. In Chapter 3 the complete characterization is given for the tensor product of two arbitrary finitely generated groups. In the last chapter we discuss the structure of A⊗B for arbitrary groups. Essentially, the only complete characterizations are for those cases where one of the two groups is torsion. Many theorems from the theory of Abelian Groups are assumed but some considered interesting are proved herein. |
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 | 2011-08-31 |
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 | 10.14288/1.0080558 |
Degree |
Master of Arts - MA |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/37036 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0080558/source |
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T H E T E N S O R P R O D U C T O F T W O A B E L I A N G R O U P S by D A V I D M I T T O N B . S c . , University of B r i t i s h Columbia, 1964 A THESIS S U B M I T T E D FN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A R T S m the Department of M A T H E M A T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A October, 1965 11 A B S T R A C T The concept of a free group is discussed first in Chapter 1 and in Chapter 2 the tensor product of two groups for which we write A ® B is defined by "factoring out" an appropriate subgroup of the free group on the Cartesian product of the two groups. The existence of a unique homomorphism h : A & B - * - H is assured by the existence of a bilinear map f : A x B - > H , where H is any group (Lemma 2 - 2 ) and this property of the tensor product is used extensively throughout the thesis. In Chapter 3 the complete characterization is given for the tensor product of two arbitrary finitely generated groups. In the last chapter we discuss the s t r u c » ture of A & B for arbitrary groups. Essential ly , the only complete characterizations are for those cases where one of the two groups is torsion. Many theorems from the theory of Abelian Groups are assumed but some considered interesting are proved herein. I l l In p resen t i ng t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re fe rence and s tudy . I f u r t h e r agree that pe r - m iss ion f o r ex tens i ve copy ing o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s * It i s understood that copying or p u b l i - c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed w i thout my w r i t t e n p e r m i s s i o n . Department o f M A T H E M A T I C S The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date October, 1965 IV T A B L E O F C O N T E N T S Chapter 1 : The Free Abelian Group Over a Set Chapter 2 : The Tensor Product of Two Groups Chapter 3 : The Tensor Product of Finitely- Generated Groups Page Chapter 4: Structure Theory to Date of A ® B for A r b i t r a r y Groups 1. Two T o r s i o n Factors — 16 2. One T o r s i o n and One T o r s i o n Free Factor 19 3. A n A r b i t r a r y Mixed Group as the Non " T o r s i o n Factor 22 4. The General Case — 25 Bibliography 27 i V N O T A T I O N The following notation wi l l be used throughout this thesis. If f : S—fe»T is a function and s feS (function and map are used interchangeably) then for the image of s under f in T we shall write sf ; for the image of S under f we write Sf. If g : T—>W is another function, then the composition of the functions f and g wi l l be written fg : S - ^ W . The symbols of set inclusion, Q and ~^ and intersection and union, C\ and U , are standard, as is the symbol of summation . The expression <̂ a> wi l l refer to the subgroup generated by the element of a group G . The isomorphism between two groups A and B wil l be written as A = B . v i A C K N O W L E D G E M E N T I wish to sincerely thank D r . R. Westwick for his assistance during the preparation and writing of this thesis. Not only did he suggest the topic for research, but also c lar i f ied and simplif ied many of the theorems used. I am indebted also to the National Research Council of Canada for financial assistance during the summer of 1965. C H A P T E R 1 ; The F r e e A b e l i a n Group O v e r a Set The subject of this paper, the tensor product of two A b e l i a n groups, involves i n t r i n s i c a l l y the concept of a f r e e group (the tensor product m our case might better be t e r m e d the "free b i l i n e a r product", as by Fuchs) and so this topic w i l l be dealt with f i r s t i n its own right. In this thesis by group we s h a l l mean A b e l i a n group throughout. L e t S be any set. We say a group G together with a mapping <y of S into G constitutes a f r e e group on S p r o v i d i n g the following condition is s a t i s f i e d : i f C^is any other mapping f r o m S into any other group H, then there exists a unique homomorphism h of G into H such that the following d i a g r a m commutes : The i n i t i a l step f r o m this definition would n a t u r a l l y be to show the existence of a f r e e group" ( G , *+* ) f o r an a r b i t r a r y set S, but f i r s t let us show that i f f r e e groups on S do exist, then any two on the same set are equal up to i s o m o r p h i s m : L e m m a 1 - 1 Any two f r e e groups on the same set S are i s o m o r p h i c , m fact, between ( G , 1 ^ ) and ( G , f' ) there exists a unique i s o m o r p h i s m i : G - ^ G such that V' ~ ^ 1 P r o o f : By our definition of ( G , H-1 ) and ( G ' ) as f r e e groups on S we have the following doubly commutative d i a g r a m : and ^ - Thus (H1 h) h' - f - ' -h ' - , but a l s o M ^ « H> , where 1^ is the identity map of G,^and we conclude h h' = I . In an entirely analagous fashion, we also find h.' h = I j , and thus h and h are both isomorphisms and h wil l fit the statement of the l e m m a , q . e. d. We shall now show the existence of a free group on an arbitrary set S by actually constructing one : Theorem 1 ° 2 A free group (G.H' ) exists for any set S. Proof : Let Z denote the ring of integers and let F(S) = \_ f | f : S —o Z and sf= 0 for al l but a finite number of seS j Define V : S F(S) by sV=f s , where t f = Sstr (Kr oenecker function) for t&S. Then first ly F(S) formsa group under the following rule of composition of functions : s(f*g) = sf + sg where sfeS, f, ge-F(S) Associativity and commutativity are t r ivial under this rule since Z forms a group. Let fft be the zero map on S to Z i . e . sf c = 0 for all sfeS ; then s(f + f j - sf + sf0 - sf for all s^S and hence f •* fo = f0-tf •* f i . e. there exists a neutral element f 0feF(S). F o r any f e F(S) define - f as follows : s(-f)= - s f for a l l seS ; then s(f-K«»f ))= sf » sf=0=sfo for all seS and hence for any feF(S) there exists an inverse element - f such that f 4 ( - f )« f . * ' A We show (F(S),V+') is free. Let H be any group and let there be defined a map ̂ : S H . If ffeF(S) define a map h : F(S) - * H a s follows : fh = ̂ j (sf)(s f } if f + f , and f„h=g , the neutral element of H . We see then that such an h satisfies the requirement s^f = sty h, and it remains to prove h is in fact a unique homomorphism. Let f ( , fjfcF(S) ; then (f,+ f t) h = 2 S s(f, + fj(s<f) ^ 5 (sf 1 + s f j ( s f ) - 2̂ ( s f j f s^ ) -+• 2 ( s f j fs^ ) by definition of composition in F(S) and by the associativity of the group H . To show h is unique .suppose there is another homomorphism h' : F(S) H> H such that H FCS) commutes. Thus s ^ * s ^ h » s^h' for al l s * S . But then for any ffeF(S) we have : f h - 2 (sf)(s<P) - S_ (sf)(sVh) = 2 s ( s f ) ( s H , h ' } - 2 ( s f H s S V S t ? , / I = fh and we conclude h=h and thus the pair (F(S),V+/) we have constructed forms a free group. We can now deduce a few easy lemmas to show the actual c o m - position of the group F(S). L e m m a 1 - 3 If (F(S),U^) is a free group on a set S, then ^ is an i n - jection. Proof : Let s f and s a be elements of S, s (* sT. Let f& : S-o Z be as before, and let h be the homomorphism of F(S) into Z produced by f. , i . e . f « = t h . Then we have s . ^ h ^ l and s7Yh--0 ; thus s,H>f s^V . . . / s ince - 4 - since h is well defined as a homomorphism, q. e. d. L e m m a 1 - 4 SH^forms a set of generators for F(S). Proof : Let F ' (S ) be the subgroup of F(S) generated by S^. Let V be the map of S-+- F ' (S ) such that sH>'= sty for all seS . Then there is an f : F(S) -o F ' (S ) such that ^ f = Let g : F ' (S)-»F(S) be the injection such that xg=x for all xfeF^S). We have then the diagram : / Then fg : F (S) -^F(S) and s^fg = sVf = s V = sV for al l sfcS. Hence fg is the identity on F(S) and therefore g is onto, and so F 7(S) = F(S), q. e . d . Recalling the construction of the unique homomorphism connected with a free group, we can now, in fact, represent any element of F(S) in terms of the generators, f c fcS v f : fh = 2 (sf)(s^) = (sfXsH'h) = S (sf)(sV) h and thus f =• (sf)(s^)= ^ (sf)(f<-). Indeed, for tfcS we have t ^ (sf)(f s) - 2 (sf)(tf^) = (tf)(tft) = tf Now suppose 2^ U ^ f ^ f ^ , ^t,feZ , f^S^V, and t t S is arbitrary ; then t 2i M4fc,= 0 ^ S r* 4(tfj = 0 Thus there exist no relations between the generators of F(S), and hence the use of the word " f r e e " to describe (F(S),H ;). A s an example of a free . . . / g r o u p - 5 - group we might cite the direct sum of an arbitrary number of infinite cyclic groups, which clearly forms a free group over the index set of the direct sum. C H A P T E R 2 : The Tensor Product of Two Groups Let A and B be two groups, let S=AXB be their Cartesian product and let the pair (F(Ay-B),*+/) be a free group over A * B . We then define the tensor product of A and B as follows. In F ( A K B ) we can consider elements of the form : (1) (a(+ at, b ^ - f a , , b)M^-(ai, b)4> (2) (a,b (+b a)H'-(a,b,)H /-(a,b l)^ where a,, â fc A and b ,b x eB. Let _C\_ denote the subgroup of F ( A * B ) generated by the set of elements of the form (1) and (2) ; we now define the Tensor Product of A and B to be the factor group F ( A * B), for which we write A<2>B. The map V : A*-B - » F ( A xB) given by the composition of the'maps : A x B - ^ F j A s B ) , .Ti- the free map, and v\ : F(A>- B) -fi" F ( A x B ) , the natural homomorphism, is called the free bilinear map, or the tensor map ; for (a,b)y we write a(g)b. Definition 2 - 1 A map "f : S ^ S ^ T from the Cartesian product of any two groups S, and S z into a third, T , (all groups having the law of composition denoted by+) is called bilinear if (s.,+ V s j f * ( s«» s J f ^ ( s . « . > ' s J f a n d (s«' su+ S M ) f* (s„ , s j f + (s,,, szi) f for a l l s ) (, sltfeS, and su, sz£.Sx. Clear ly the tensor map <̂ : A * B - * » A ® B is bilinear since by, (1) ( a ^ a ^ . b ) ^ -(a,,b)V - ( a t , b ) ^ - 0 (or (a(+ a j ® b « a , ® b + a , ® b) and by (2) (a, b,-t-H> -(a,b,)H> - ( a . b J ^ ^ O (or a65(b(+ b 2 ) = a ® h + a ® b 1 ) . - 6 - L e m m a 2 - 2 Let f : A X - B - ^ H be a bilinear map from the Cartesian product of two groups A and B to a group H . Then if '• A ^ B - ^ A ^ B is the tensor map, there exists a unique homomorphism h : A C ^ B - ^ H such that the following diagram commutes : Proof : Let ( F ( A * B ) ,H>) be a free group on A * B and let h / : F(AX-B)—*>H be the unique homomorphism associated with (F(Av:B),H J) and the map f ; v 1 „ : = Then, f being bil inear , we have (a-» a', b ) f - ( a , b)f - (a', b)f=0 and (a, b+b' )f -(a, b)f -(a, b' )f=0 for al l a, a ' e A and b,b'feB , hence (a+a'.bjM'h' - (a .b)^h ' -(a' ,b)^h'=0 and (a,b4-b' ) f h ' - (a .b )^h ' -(a,b ')M^h'-0 which implies (ker h') ~2 - O - ; therefore, if we define, for f f rF(AxB) , (f +• £X)h = fh' , where f +T1 =fwj the coset of f in A6? B', we have asserted the existence of a homomorphism h which fits the lemma. The uniqueness of such an h follows from the requirement that the diagram in the lemma should commute, q. e. d. It is clear f rom the properties of the free group (Lemma 1-1) that the tensor product A ® B is unique up to isomorphism ; we can now also prove the following : Lemma 2 « 3 A ® B@ A for any groups A and B. Proof ; Define the following maps : f : AxB—*»B*-A and g : Bx.A-»A*B by (a, b)f=(b, a) and (b,a)g=(a,b) Let V : A*B - s > A ® B and : B<A-*- B(*> A be tensor maps and we therefore have the following diagram : i A <0 A *4> f ' and g' are homomorphisms whose existences are assured by the bilinearity of the compositions f and g <y . Now fg is the identity map of A*B , hence f'g' is the identity of A ® B ; similarly g f is the identity of B(g> A, thus A<$B^=B®A , q. e. d. The following will give us now a set of generators for the tensor product of two groups, the sets of generators of the latter being known : Lemma 2 •» 4 If A 0 is a set of generators for A and B 0 a set for B, then ©̂<<?>p, \ <xe A 0 and (It B0̂ forms a set of generators for A ® B. Proof : -We know already (page 4) that any element of F(AxB) may be written as 2J "\ (av , b ) 4*, where \ t €. Z and (a^, b^V^A^BjM-' , and at the same time the set ^ = ̂ (a, b)V ja feA , bfe BJgenerates F(A*B). Then since ̂ , the natural map, is onto , ̂ (a,b)¥*j \ a€A, bfeB^a©b[afcA,bfcB^ . . . /generates generates A&B. Now a e A implies a. = ~2s "C^ , " s ^ Z and a t A . ; s imilar ly b e B implies b = 1» "S, b. , S fcZ and b e B . . Thus the bilinearity of the tensor product implies We can now prove some easy lemmas based mainly on the bilinearity intrinsic to the tensor product. L e m m a 2 - 5 (a) If either a-0 or b = 0, then a®b= 0, the neutral element of the group A ® B . Proof : Assume a = 0 ; then a^a-v-a and a©b = ( a + a ) ® b = a(g)b-t-a®b , hence a®b= 0. S imilar ly b= 0 implies a ® b - 0 , q. e. d. 2 » 5 (b) (•a)cg>b = » ( a ® b ) =a(S>(~b) for any a e A , bfcB. Proof : ( « a ) @ b ^ a<g>b = (a»a)<2>b= 0 , thus (»a)<g>b= »(a<©b) ; s i m i l a r l y a0(»b)=«»(a<3 b) , q. e . d . Corol lary : a ® b = ( » a ) ® ( » b ) 2 a 5 (c) n a ® b - n ( a ® b) = a 0 n b for any n t Z , a t A and b t B . Proof ; By induction. C H A P T E R 3 : The Tensor Product of Finitely Generated Groups In this chapter we shall determine the structure of the tensor product of two finitely generated groups. The concept of a direct sum of Abelian groups wi l l be used extensively and therefore a workable definition wi l l f i rs t be formulated. Definition 3 » 1 Let H be a set and for e a c h ^ e H , let B^ be a group. By the direct sum of the B^ , for which we write ^ B^ or BfcfeB.,© >" B^tf)*-1 , we mean the following : - 9 - 2 , 1 ' B 0 t = [ f : H - v U B * \ o c f f e B ^ a n d <* f f 0 f o r a t m o s t f i n i t e l y m a n y w h e r e a d d i t i o n o f e l e m e n t s i s p e r f o r m e d as f o l l o w s : l e t f . a n d f, £- B . , ; t h e n f, + f ' * Ken " i 2. d e f i n e d b y . <X (f (+ f 2 ) = oc f ( + o< f 2 . H ^ O B . i s odfcH T h i s d e f i n i t i o n c o v e r s t h e i n t e r n a l c o n c e p t of a d i r e c t s u m m w h i c h a g r o u p G i s s a i d . t o be t h e d i r e c t s u m of s u b g r o u p s B ^ i f g e G i m p l i e s K g = 2, b- t w h e r e b^e B c a n d s u c h a r e p r e s e n t a t i o n i s u n i q u e . T h e f i r s t m a i n t h e o r e m , w h i c h w e s h a l l u s e e x t e n s i v e l y t h r o u g h o u t t h i s p a p e r , i s o f i n t e r e s t m i t s o w n r i g h t : T h e o r e m 3 - 2 L e t A be a .ny g r o u p a n d l e t B = 2 , ' B ^ be a d i r e c t s u m o/fert t h e n A ® B - 2.* ( A <® B j P r o o f : C o n s i d e r t h e f o l l o w i n g d i a g r a m PUBv «=- ft 4 9 9, I5 ft® B » 1* -o- A© Bp w h e r e B » , - _ B y a r e d i r e c t s u m m a n d s o f 2., B a n d t h e m a p s , H 3 ^ » ^ a r e t h e a p p r o p r i a t e t e n s o r m a p s . T h e m a p s l ^ a n d p ^ a r e d e f i n e d a s f o l l o w s : - 10 - \ A : A K B -*>A *. 2' B̂ . where (a, b)L= (a, f«) and j ^ f ^ b ? f o r a l l a t A , b e B „ . U fp = 0 i f ot*|& J > C> : A ^ 1\'B^-«> A^B-u where (a, f)Pjf= (a, tff) for a l l afeA and f fe- 2?V*B... It is c l e a r then that the compositions l . ^ and OC6H 1 a r e b i l i n e a r so the maps j^^ andv^ indi c a t e d on the d i a g r a m do exist and are i n fact homomorphisms, so that \{V = Vppp. and p8vpy = H> "|* . Thus (a® b ) ^ = a®f(j for afe A and b t B 3 and also (ag) f W = a © b for f <£• ^'Bu a r b i t r a r y except that |3f =b. Next c o n s i d e r the map h : %'(A® Bu>-t>A <g> 2,' B.defined as follows : let f£-2'(A® Bj) be an element such that ^ f ^ y ^ ^ A * Bp, for a l l fe H ; then let fh = "^Y-pJup , where |J ranges over those elements of H such that ^ f tO ; the sum i s thus f i n i t e . ^N. B. h i s a h o momorphism ; the fact that (f -t-f2) h= f(h-v-f,_h is c l e a r f r o m the definition, as is the fact that 0h= o). Now it is evident that A ® 2<* B^ is generated by : \ ag) f I afeA and f £ 2, B,/ where f has the f o r m : f o r some ftc-H a n d b e B . «! f = b i f <*• = /> for a l l &d e H { . Thus h is onto = 0 if<*tp ^ A ® 2j*B^ , so for the d e s i r e d i s o m o r p h i s m we need only show k e r (h)= 0. Suppose, then, that %h^~ 0 — 5} ; b u t ^ i s a h o m omorphism for any \ 6 H, and thus ( 2\ Xp^p) also. However, we know f r o m c o n s t r u c t i o n that : [ identity of A * B^ i f p>= * \_ annihilator of B^ component if p>t* ( identity of A ® Bp i f rt=* and thus AA* " V - I y ' (^zero map of A & i f f*>*x - 11 - Therefore ( 2* j u f )r\* = 0 •=• 1* and we conclude h is an isomorphism , q. e. d. C o r o l l a r y 3 - 3 2,'A* (g> 2,'B(> - 2,'(A^® B.) Oifc*\ fJfcl <*<-« Now let G be a finitely generated group ; then it is well known that G = T © F where T is the torsion subgroup of G and F is torsion free. With this information, then, we know that : Gy®G2 = ( T © F ) & ( T V © F 2 ) = ( T , ® , r i ) © d \ ® F J © ( F ® T ) @ (F,(g)FJ where G, , G r are finitely generated groups and G^= Tt<i) F v describes the decomposition into torsion and torsion free summands. To completely describe the tensor product we must analyze each of the four direct summands above ; the second and third are essentially the same ( A $ B = B ® A ) so that it remains to examine three classes of tensor products. The first class is merely the restricted case where finitely generated is replaced with finite, s i n c e finitely generated torsion and finite are synonomous. Definition 3 <* 4 A torsion group T is called a p - group if every element has order p for s o m e o t t Z , where p i s a p r i m e . There is the basic theorem for the decomposition of an arbitrary torsion group into its p - subgroups as follows : Theorem 3 - 5 A torsion group T is isomorphic to the direct sum of its p « subgroups, where p ranges over all p r i m e s . Proof : Let T p = ^ t & T | t has order p* , some^fe z|. Then T ^ t T is a subgroup , for if a,b«iTp then ]? a = p b= 0 for someoi , AfcZ whence p'v,KxCo<>^ ( a-b) =0. Now T p O 2 Tq = 0 since any element of this intersection must have orders p and p , 12 - two relatively prime numbers and the neutral element is the —j only such element. Thus the sum ^, T p is direct and to P complete theorem 3 - 5 we need the following : Lemma 3 - 6 Let x €. T have order n=n(-nx->" - nowhere the n t are relatively prime in pairs ; then X has a representation as x:x^ x2+.MXte where (order x;) = n-t , Proof : We prove the lemma for the case k= 2 , the induction to general k being easy. We then know there exist integers a, b such that an,-tbna=l and hence x « an.x-t-bn̂ x . If we let x,= bn,x and xt=an,x then it is easily seen that (order x( n,' and (order x j = n^ , where n| | n( and n'l|n1 , (s 11 means, as usual, t = ss, for some s, ) and also that x= x (t Xj_ . But n'nt x = njn̂ x,-̂ n'n'x x= 0 which implies n(nr| n[n'x , and thus n,= n,' and n2=. n[_ , q. e. d. Now any x t T must have order n-p*' • p**- ••• • p*" where k,ateZ so by the lemma there exist x ( , , xlt*-T such that x = 1 x-t , x tt Tp̂ , and the theorem is proved. it Let T, and Ti be as before ; then Tt<© T v = 21' T p <£> 2, T£ where each Tp and T^ is a prime p -group and q - group respectively. Hence by corollary 3 - 3 T,® Tt = S " (Tp (s> T^ ). The following lemma will enable us to eliminate many of the cross tensor products in this expression : Lemma 3 - 7 If A is a p - group and B is a q - group for pf q primes , then A ® B = 0. Proof ; Consider a generator of A& B of the form a<g>b,any at A , b *B . Since A ® B is generated by\ a® b I at'A, b«:B^ - 13 it suffices to show that a $ b — 0 ; but p** a = q b = 0 f o r some oi , f^e Z and since p* and q ? have g. c. d. equal to 1 , there exist integers s, t such that s p % tq'' = 1 . Then a ® b = (sp"-+ tq") ( a ® b ) = ( s p % tq'*) a ® b * tq Pa $ b •= a 0 tq b = 0 , q. e.d. T h e r e f o r e , i n the tensor product (Tp' (g> T^e.) - T(<g> T, we need c o n s i d e r only those summands of the f o r m Tp(g5 T* f o r the same p r i m e p . Now each finite p «» group may be written as a d i r e c t sum of c y c l i c groups of o r d e r p " , say T^ = 2} , 1=1, 2 , and thus Tp4S> T* = ®2,*C^ = Z ' ( c ^ ® Cj«). We s h a l l now prove that any summand of the l a t t e r can be fu r t h e r s i m p l i f i e d : L e m m a 3 - 8 C ^ o ® Cp* — C p ^ t j i v ] where C p» a r e c y c l i c groups of or d e r p* , p p r i m e . P r o o f ; L e t C^p. - <A>and Cp« = <b>and assume without l o s s of g e n e r a l i t y that ^3< . C o n s i d e r the following d i a g r a m : where f : Cpf»x Cp* C^p i s defined by (na, mb)f = nma , n, m feZ . Th e n (1) f i s w e l l defined : suppose n v= n (p p) and m ( H m ((="*) i . e. n(= n+ and m = m t-tp f o r some s, t e=Z. - 14 - Then (n,a, m,b)f = n v m ( a= (n-t-spP)(m-t- tp) a = (nm+ ntp*+ smp ? + stp^p*) a •= nma (p )̂ N . B . If we had defined f 1 : C pfi*C p* — C p < in the same fashion , the map f ' w o u l d not be well defined here! (2) f is bilinear : (na-vn'a, mb)f ((n+n') a, mb)f = (n+n') m a - nma V n 'ma - (na, mb)ff (n'a,mb)f S i m i l a r l y , (na, mb-vm' b)f = (na, mb)f +(na, m'b) f and thus the homomorphism h : C p P ® Cp» C p(* exists. We must show now (1) h is onto : clear , since (na, b)h=na , a general element of C . (2) h is one to one : Let 2» (n a<g>m;b) represent an arbitrary element of C^p® ; but then n a ® m b = ) t a ( ^ n m b t a ® J n m. b _ _. 1 * t • - . V. \, ^—1 I i * - \ w t i _ N a i ^ b , some Ne Z Suppose therefore (Nag>b)h= 0 =• Na ; hence N = 0 (p )̂ and N a ® b - 0 , and h is an isomorphism, q. e. d. We finally have, then, T((£> T, = ^ * T p <g> 5/T^ 2f 2,'(T;«> T^M = 2,'(Tp <a T ; ) = S'tc^® c p\) — ^ C p « l , e. a finite direct sum of cyclic groups of prime power order . The second class of tensor product to examine is of the form ' ^ ' ^3 • Now any finitely generated torsion free group . . . /may be " 15 *» may be represented as a finite direct sum of copies of the integers, and thus T_© F, = Tt<£>2i*Z = 2^'(T;_®z). We need at this point the following lemma : L e m m a 3 * 9 G <£> Z = G for any group G . Proof : We may again prove this by the basic lifting property of the tensor product ; consider the following diagram : Gr*. Z» A 4> A A. G K 2 where f ( : G x Z — G IS defined by (g ,n)f t =ng and f x : G-*>G*-Z by gfj.-=-(g, 1) , Mf" is the tensor map and I : G - * G is the identity map. Now the existence of A A the homomorphisms f x, f, is assured by the bilinearity of the composites f, I and f ^ f . Let (g,n)€rG*Z ; then A. A A A. (g, TL\)ixf*{ngtl) so that ( g ® n ) f,fj=ng® 1= g<S>n. Hence f, f 4 A A is the identity of G ® Z and s imilar ly f^f x the identity of G , which shows G ® Z = G , q. e. d. We have now, then, that T-t (& Fj £ T v ® 2^'Z $ 5/Cr f c<3 z) The final class was typified by F (g> F^ , the tensor product of two finitely generated torsion free groups, which can now easily be ex« pressed as ^ ' Z ®2,*Z - 16 - = :?,'(z® z) 2,'z We have thus shown that the tensor product of two finitely gen- erated groups can be expressed as a direct sum of cyclic groups. The latter may be explicitly calculated by determining the decomposition factors of the two given groups. C H A P T E R 4 : Structure Theory to Date of A(g)B for A r b i t r a r y Groups We now turn to the examination of the structure of the tensor product of two groups at least one of which is a torsion "group (not necessarily finitely generated). 1. Let us first deal with the case of two torsion groups. By C o r o l l a r y 3 - 3 and Theorem 3 - 5 , there is no restriction of gen- erality in considering the tensor product of two p » groups, and, m fact, P * groups for the same prime p , by L e m m a 3 - 7 • However, we are not restricting ourselves to finitely generated groups, so that we may not, in general, represent them as direct sums of cyclic groups. To c i r c u m - vent this problem we introduce the concept of a basic subgroup which is , amongst other things, the direct sum of cyclic groups, and prove event» ually that the tensor product of two p - groups is essentially the tensor product of their respective basic subgroups. Definition 4 - 1 A subgroup G 0 of a group G is said to be a pure subgroup if x & G 0 and x = nx, for some x ^ G and n<cZ , then x r n x a for some xDe G c ; symbolically we may express this by the equation nGQ= G Q R n G . - 17 - Definition 4 » 2 A group G is divisible if for every xe G and nfeZ , x = nxh for some x K & G . Definition 4 - 3 A subgroup B of G is said to be a BASIC subgroup if the following three conditions are satisfied : (1) B is a direct sum of cyclic groups. (2) B is a pure subgroup of G . (3) The factor group G / B is divisible . A fundamental theorem for basic subgroups of p » groups is the following of Kulikov ; the proof wi l l only be sketched here and it may be found in detail in F u c h s , Abel ian Groups, Chapter 5. Theorem 4 » 4 Every p « group contains a basic subgroup. Proof : A pure independent subset £ x x ^ ̂ > that is an independent subset which generates a pure subgroup of G , exists and may be extended to a maximal set in G by Zorn,'s l e m m a . Then if B is the subgroup generated by the maximal pure independent set ^ K f c j \ _ ' ^ * n t ^ i e a ^ o v e definition is true and (1) is an easy consequence of independence , (3) requires the maximality of the independent s e t ^ x x ^ ^ . Now if A , and B^ are subgroups of A and B respectively then it is not generally true that A,(£) B ( forms a subgroup of A(£>B but merely that the subgroup of A<§)B generated b y ^ a ( & b | a f c A ( , b e B , ^ is a homo* morphic image of A,(£} B, . If, however, A , and B, are pure subgroups we have the following : Theorem 4 - 5 If A , and B f are pure subgroups of A and B respect- ively then the subgroup 01 of A ® B generated by the set \ a ® b|aeA, , beB, is isomorphic to A ( ® B , . » 18 « P r o o f ; C o n s i d e r the diagram : where (a,, b ( )f = a, (g> b, , then h is a homomorphism such that f = H> h . Thus we need to show that i f an element a t ® b-t , a.e. A, , b t e B, , of A ® B vanishes then it also vanishes as an element of A , ® B,. We need the following : L e m m a 4 » 6 If 2» a^gjb^ 0 m AC&B then there exist r> f i n i t e l y generated subgroups A * and B * such that 2, a tg>b t vanishes as an element of A * ® B * . P r o o f : 2> a t ® b t = 0 only i f 5, (a , bJH^ belongs — — — — — >.i. » to the subgroup X I . generated by elements of the f o r m (1) (a,-e a ^ b j y - ( a „ b ) V -K,b ) M > and (2) (a,b,+b»)^ - ( a . b , ) ^ - ( a , b 2 ) ^ . Define A * to be the subgroup of A generated by a,, a z, — , a„ and a l l a 0 o c c u r r i n g m the e x p r e s s i o n of (a-,bt)H^ by means of elements of the f o r m s (1) and (2) above. D e s c r i b i n g B * s i m i l a r l y , we have the d e s i r e d r e s u l t s . We can now see that ^ at<g>bv vanishes as an element of A N A. A ® B where A is that subgroup of A generated by the pure subgroup A, and the fi n i t e l y generated subgroup A * (B is defined analagously) ; but it i s known that since A * and B * are f i n i t e l y generated and A, and B ( a r e pure, the la t t e r are A A d i r e c t summands of A and B and hence by C o r o l l a r y 3 » 3 *\ 2i a t ® b L vanishes as an element of A(<g) B, , q. e. d. - 19 - T h u s A B , m a y be c o n s i d e r e d a s u b g r o u p o f A ® B a n d t h i s . p r e p a r e s us t o p r o v e t h e f o l l o w i n g : T h e o r e m 4 - 7 If A , a n d B , a r e b a s i c s u b g r o u p s o f t h e p - g r o u p s A a n d B, t h e n A ( ® B, = A<8>B . P r o o f : S i n c e A , ® B ( C A ® B w e n e e d o n l y s h o w a n y e l e m e n t o f A ® B o f t h e f o r m a<8>b b e l o n g s t o A , ® B , . N o w a.e A i m p l i e s a = a,-v p x w h e r e a , e A , , k<=Z a n d x ^ A , s i n c e A / A , d i v i s i b l e i m p l i e s a = p t e x ( m o d A , ) ; s i m i l a r l y b = b - v p ^ y , w h e r e b ^ B , , , y€ B . N o w c h o o s i n g p* "2 ( o r d e r b) a n d p* Z. ( o r d e r a,) w e h a v e a <g> b = ( a , + p h x ) ® b = a , ® b = a, €> (b,-*-p*y) = a, <g> b , T h i s s h o w s A ® B ^ A , © B ( a n d t h e p r o o f i s c o m p l e t e . W e a r e n o w m a p o s i t i o n t o p r o v e t h e m a m t h e o r e m o f t e n s o r p r o d u c t s f o r t w o t o r s i o n g r o u p s : T h e o r e m 4 - 8 T h e t e n s o r p r o d u c t o f t w o t o r s i o n g r o u p s i s a d i r e c t s u m o f p r i m e p o w e r o r d e r c y c l i c g r o u p s . P r o o f : B y t h e l a s t t h e o r e m i t s u f f i c e s t o p r o v e t h e p r o p o s i t i o n f o r t h e t e n s o r p r o d u c t o f t h e b a s i c s u b g r o u p s o f t w o p - g r o u p s , b u t t h e s e a r e d i r e c t s u m s o f c y c l i c g r o u p s a n d t h u s b y C o r o l l a r y 3 - 3 a n d L e m m a 3 - 8 t h e s t a t e m e n t o f t h e t h e o r e m i s t r u e . 2 . T h e c a s e w h e n o n e o f t h e f a c t o r s i s a t o r s i o n g r o u p a n d t h e o t h e r i s t o r s i o n f r e e w e s h a l l e x a m i n e n o w . A g a i n w e m a y a s s u m e t h e t o r s i o n g r o u p T i s a p - g r o u p , a n d l e t F be a n y t o r s i o n f r e e g r o u p , i . e . f o r a n y g f c F , n g = 0 i m p l i e s n = 0 , n e . Z . C h o o s e a m a x i m a l i n d e p e n d e n t s e t m F m o d u l o p F , s a y ^ x ^ ^ ^ L e m m a 4 - 9 S u c h a s e t { . f o r m s a b a s i s o f F m o d u l o p * F v& A. n - l , 2 , 3 , - - • - 20 - P r o o f : F i r s t i f | x v | i s a m a x i m a l independent set m F / p F then for any x f c F , x ^ p F we must have mx^n ix,+ n zx z+ "• + n t x t - 0 (mod pF) f o r m,n t& Z not a l l z e r o . Now p|m i m p l i e s ^ n t x , - 0 (mod pF) and not a l l n-̂ z e r o , which contradicts the independence of the x t , hence ( p j i r t j ^ l and there exist integers a,b such that ap+bm-1. But bmx = n'x,+ ••. + n'x. (mod and bmx = (l-ap)x = x (mod pF) and thus x = n'x,-*- - + nJ*x (mod p F ) . C o n s i d e r an element y t p F , y ^ p F. We have then that y=p f, f<=F but ff-pF and thus f = 5, n (mod pF) which i m p l i e s y= 2, m x x «•»! (mod p F ) . Note a l s o that if an element y of F has finite height k \ i Si n then y has a rep r e s e n t a t i o n y = 2̂ m v x v (mod p F) f o r arbitraryJ(.>k, for let y s m x (mod p**F) be the representation a l r e a d y proved. Then suppose , without loss of g e n e r a l i t y , that y - !?i m x ^ p * F , But then (y - 2 i rn^xj - ep * F for some n^e Z, thus y - ~2A =• 0 (mod p F) and i n gen e r a l the congruence y - "2. m^x^ = 0 (mod p F) is solvable f o r arbitrary£>k, q.e.d. C o n s i d e r now any t & T and any x ^ F of height k. Note that if f€ F has infinite height ( i . e . p"J f f o r e v e r y nfe Z) then t<S>f = 0. T h e r e exists the r e l a t i o n x= n x,t - + nsxs-*- p f f o r x L& ^ x v ] and a r b i t r a r i l y l a r g e *<£ Z , choosing p i ( o r d e r t) we obtain : t&x = 1, t O n x + t ® p f = 2 t t ® x t where t t = n^t We see t h e r e f o r e that an a r b i t r a r y element of T ® F , being a finite sum of generators of the f o r m t®x , may be written a l s o as a finite sum 2i t 4 « x t f o r t,<? T a n d d i f f e r e n t x s e l e c t e d f r o m t h e i n d e p e n d e n t s e t 1 x , v > s i n c e a d d i t i o n i n t h e g e n e r a t o r s m a y b e c a r r i e d ou t o n t h e t € T . S u p p o s e n e x t t h a t a s u m s u c h a s % t k ® x t v a n i s h e s a s a n e l e m e n t o f T ® F . W e s h a l l s h o w t h a t t h i s i m p l i e s t t = 0 , i « 1, , n , a n d a g e n e r a l e l e m e n t o f T ® F m a y b e e x p r e s s e d a s a n e l e m e n t o f a d i r e c t s u m of c o p i e s o f T . B y L e m m a 4 - 6 1 t t ® x 4 v a n i s h e s a l s o a s a n e l e m e n t o f T <2) F ( w h e r e F , i s f i n i t e l y g e n e r a t e d t o r s i o n f r e e a n d c o n t a i n s x ( , • • • , x H ; b u t t h i s m e a n s F , i s a f i n i t e d i r e c t s u m o f i n f i n i t e c y c l i c g r o u p s g r e a t e r t h a n o r e q u a l t o n i n n u m b e r . H o w e v e r , x ( , •<> , x „ b e i n g i n d e p e n d e n t , F , m u s t c o n t a i n a d i r e c t s u m m a n d F ' c o n t a i n i n g x , , . x ^ o f r a n k n a n d t h e n 2, t t ® x v m u s t v a n i s h a l s o i n T ® F ( ( T h e o r e m 3 - 2 ) . S u p p o s e , t h e n , *\ I —\ F ( s < a,> (£> < a x > © " * © ^ a K > . T h i s m e a n s x ; =2, m a 4 a n d t h u s X t t © x t = 2, t 6?2 m a = S, ( 2 . m t v ® a ) = 0 h o l d m t h e t e n s o r n p r o d u c t T <g>F( w h i c h i m p l i e s 2̂ m ^ t t ® a , = 0 ; t h i s i n t u r n , b y L e m m a 3 - 9 i m p l i e s 51 r n , t = 0 . L e t A b e t h e m a t r i x ( r n ) ( . I f A i s a s i n g u l a r m a t r i x ( m o d p) t h e n j -i - »> L i e s t h e r e e x i s t y3;,€Z n o t a l l z e r o s u c h t h a t 2 | i t m t J = 0 ( m o d p ) , a n d t h e r e f o r e S, (̂ ^ m t . ) a j = 0 ( m o d p ) . B u t J (£ /*• m. ) a , = ̂ d j m = I A x , *• a n d t h u s ^ |3V x t = 0 ( m o d p ) . T h i s , h o w e v e r , c o n t r a d i c t s t h e i n d e p e n d e n c e of t h e x t , a n d h e n c e (p , d et A ) = 1. L e t d e t A = K a n d t h e m a t r i x ( ) t , M be d e f i n e d a s K • A . N o w m v t t * 0 f o r e a c h j = 1 , ' • • , n r m p l i 5 i rn^ t = 0 f o r e a c h k - 1 , ••• , n w h i c h i n t u r n i m p l i e s 5 , ^ 1 m ^ t t « 0 % (K S t l t ) t t = K t ^ = 0 w h i c h i m p l i e s t^ =• 0 . - 22 - We have thus p r o v e d the following : T h e o r e m 4 » 10 If T is a p - group and F i s t o r s i o n f r e e , then T(2>F = X\* T , where m denotes the rank of the factor group F / p F 3 L e t us now con s i d e r the case when the non»torsion f a c t o r is an a r b i t r a r y m i x e d group M. We may assume the t o r s i o n group T i s a p e group, and by the following l e m m a it is sufficient to assume the t o r s i o n subgroup of M i s a p » group f o r the same p r i m e p. L e m m a 4 - 11. L e t A 0 and B^ be subgroups of A and B r e s p e c t i v e l y , then A ^ B, 2r A © B where P(A0, B 0 ) 9 A ® B is generated by \ m ® n I meA nor V B , " r ( A 0 ) B j V U6BO] P r o o f : L e t ei : A ® B — A ® B be the n a t u r a l homomorphism. ' r ( A 0 , B o ) Then (a, b)f = (a®b)iq is a b i l i n e a r map vanishing whenever afe A c or b f c B D ; this means f depends only on A and B̂ _ , A„ Be I. e. f : B^ _ j , A ® B is b i l i n e a r , c l e a r l y A „ X B 0 T ( A 0 , B Q) ^(a€>b)w a e a e A , b e b & B ~l generates A<S>B and thus there 1 A „ BoJ WA..B A c 0 ) r ( A 0 , B 0 ) Y(Ao,B0) exists a homomorphism h f r o m A ̂ B_ onto A ta) B i. e . (a®b)rj = (a,b)H>h. Now ( ̂ ( a t , b J S ) ) h = 0 only i f ̂ a ® b e ? ( A t , B 0 ) i . e . 2, A<® B T belongs to the subgroup of A ® B generated by a l l a ® b where either "a - 0 o r b - 0 N. B, F o r x e A x denotes the coset x+ A 0 , then c e r t a i n l y 51 (at> D t)V = 0 a n d n 1 S indeed an i s o m o r p h i s m q. e.d. - 23 Th i s lemma shows, using A = T , A o = [o] , B - M and B G = ( sum of q - subgroups of M f o r a l l p r i m e s q t p ) , that T <S> M/B 0 = T<8M ,since r (A o,B 0) = {o^ , and M/B Q has only a p - group as t o r s i o n subgroup. We now state and prove the m a m theo r e m of this s e c t i o n : T h e o r e m 4 - 1 2 Let T be a p - group and M a mixed group whose t o r - sion subgroup M Q is a p - group. Let B be a basic subgroup of T which is represented by 2 ^ C(p L) where C(p l) are p r i m e power o r d e r c y c l i c groups, Then T ® M = T 2" T / p l T © % ' T .where f denotes the rank of J^T\ P r o o f : Let { a - x j x e j ^ be a basis of the b a s i c subgroup B and (.x^J^^ be a m a x i m a l independent set m ' ̂ ^ e n \ a>>^ is a basis of T/p T for e v e r y k > l , since T/"B d i v i s i b l e i m p l i e s t = p t + b for any k £ Z and some be B ; also) x^C is a basis of ?̂<M",\ f o r e v e r y k2. 1 as we know f r o m \ 2. Hence if x, e x„ •p*C%' • » / • / * is an a r b i t r a r y element f o r allym€. N , then the set ^ a x , x ^ | f o r m s a basis f o r M/p M f o r a l l k ~z. 1 . A g a i n , as in J 2 , we need only c o n s i d e r elements of M of finite height k , so i f v t M is such an element there exists the equation : ^ ^ fi / rt v= S m a . + 2 n x, + p v', where m , n <=• Z and X > k is a r b i t r a r y x»> v x /* r ^ r m Z, and v € M . Choosing ^ a p p r o p r i a t e l y large , we have : t <5> v = 51 t ® m v a x + 2. t<g> n^x^ + t ® p*v' = 2 t x® a v + 2, t^fc* x^ Thus again an a r b i t r a r y element of T ® M may be written as 2, t x ® a y +2, t^Sx^, since addition may be c a r r i e d out on the t x and , using the b i l i n e a r i t y of the tensor product. We must show that such an element of T ® > M may be wri t t e n as an element of a d i r e c t sum, i . e . i f X t ® a. i« t ® x = 0 then each summand vanishes a l s o . But any finite d i r e c t summand of B , < a , > © ••• ©<a > x>, is a d i r e c t summand a l s o of M since - 24 - BSM,.say M= <a,> © - - - © < a*> © M'; by the choice of the [ av^ and \ x ^ , M maybe assumed to contain x( , ,xh. Therefore by Lemma 3 - 2 each ty& a v - 0 and from § 2 t̂ ® x^- 0 and thus tp- 0 for eachyy. . The above shows that T O M — T,'(T<S> <av> )© 2i"T . We give the following lemma which completes the proof of Theorem 4 - 12 , since the <a y> are finite cyclic groups : Lemma 4 - 1 3 If C(n) represents a cyclic group of order n' and V is any group, then C(n) ® V == V/nV , Proof : Consider the diagram Let C(n) = <a> where f is defined as follows : (ka,v)f = kv + nV for kfcZ , vfcV . Then f is clearly bilinear so that the map h exists and is a homomorphism satisfying f = V h . h is onto all of V/nV , for if v+ nV is a general element of V/nV , then (a ® v)h = v+ nV ; h is one to one for if 2i k ta®v c represents an element of C(n)®V such that ( X k La® v t)h = 0 , then (a® v')h = 0 (where v'= 21 ktvt fcV ) which implies (a, v') ty h =• 0 =• (a ,v')f = v' +- nV ; this m turn implies v t nV , and thus a© v'= 0 and the isomorphism is established. We can now state a corollary of Theorem 4- 12 based on the information derived m § 1 and § 2 : 25 » Corollary 4-14 If T is a torsion group and M a mixed group whose torsion subgroup is denoted by M0, then T ® M = T ® M ^ ® T ® M 0 4 In the general case, when we consider the tensor product of two arbitrary groups, very little can be said. We can,however, determine the structure of the torsion subgroup of the tensor product of two arbitrary groups M and N with the knowledge of Lemma 4 - 11. Let M0and N 0 be the torsion subgroups of M and N respectively. Now the fact that the subgroup P(M0,N9) in M ® N is a torsion subgroup is clear ; the following lemma will show that P (M0,No) is the maximal torsion subgroup of M ® N : Lemma 4-15 AflS>B is torsion free if both A and B are. Proof : The proof is clear when it is noted that if a generator a®b = 0, a6A, beB , in A6&B then a<$b-0 also in A (®B ( where A ( and Bf are finitely generated ; the decomposition of finitely generated torsion free groups into direct sums of infinite cyclic groups and Lemma 3-9 are used for the final result. We note next that M Q and N 0are pure subgroups of M and N and thus, as in Theorem 4-5, M^N and M ® N 0 form subgroups of M ® N ; they clearly generate P (M0,No). By the decomposition of M o and N 0 into their p-summands we also see that the p-component of T (M0,N0) is generated by the subgroups M 0 ® N and M$N D ?of M <8N. It suffices to consider the case, then, when the torsion subgroups of M and N are p-groups for the same prime p. The theorem is as follows : Theorem 4-16 Let M 0and N0, the torsion subgroups of two arbitrary groups M and N, be p-groups for the same prime p ; let B = 1+' C(p') - 26 - be a b a s i c subgroup of N D and denote the ranks of by d and r e s p e c t i v e l y . Then the m a x i m a l t o r s i o n subgroup of M ® N is i s o m o r p h i c to J.' MVp 1 M 0 © T , M 0 © 2 ' N O P r o o f : The proof p a r a l l e l s that of T h e o r e m 4 - 1 2 so w i l l not be given h e r e . It w i l l be noted that the t o r s i o n subgroup of M ® N is generated by tensor products of the exact type dealt with in § 3. T h i s theorem obviously t e l l s nothing of the tensor product of two a r b i t r a r y t o r s i o n f r e e groups , and, indeed, v e r y li t t l e of the nature of this type of tensor product is known to date. - 27- BIBLIOGRAPHY 1. L. Fuchs , Abelian Groups , Budapest, Hungarian Academy of Sciences , 1958 2. H. "Whitney , The Tensor Product of Abelian Groups , Duke Mathematical Journal , Vol. 4 , 1938 , pp. 495-52 8 3. L. Fuchs , Notes on Abelian Groups, I , Budapest University Annales, T. 1-4, pp. 6-14
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