Infinite Abelian Groups

INFINITE ABELIAN GROUPS Volume Z

This is a volume in PURE AND APPLIED MATHEMATICS

A Series of Monographs and Textbooks Editors: SAMUEL EILENBERG AND HYMAN BASS

A list of recent titles in this series appears at the end of this volume

INFINITE ABELIAN GROUPS Ld.szlo Fuchs Ttclane University

New Orleans, Louisiana

VOLUME I

ACADEMIC PRESS

New Yorh Sail Frawiscv Loizdon

A Subsidiary of Harcoitrt Brace Jovanovich, Publishers

1970

COPYRIGHT @ 1970, BY ACADEMIC PRESS,JNC. ALL RIGHTS RESERVED NO PART OF THlS ROOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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PREFACE

The theory of abelian groups is a branch of algebra which deals with commutative groups. Curiously enough, it is rather independent of general group theory: its basic ideas and methods bear only a slight resemblance to the noncommutative case, and there are reasons to believe that no other condition on groups is more decisive for the group structure than commutativity. The present book is devoted to the theory of abelian groups. The study of abelian groups may be recommended for two principal reasons: in the first place, because of the beauty of the results which include some of the best examples of what is called algebraic structure theory; in the second place, it is one of the principal motives of new research in module theory (e.g., for every particular theorem on abelian groups one can ask over what rings the same result holds) and there are other areas of mathematics in which extensive use of abelian group theory might be very fruitful (structure of homology groups, etc.). It was the author’s original intention to write a second edition of his book “Abelian Groups” (Budapest, 1958). However, it soon became evident that in the last decade the theory of abelian groups has moved too rapidly for a mere revised edition, and consequently, a completely new book has been written which reflects the new aspects of the theory. Some topics (lattice of subgroups, direct decompositions into subsets, etc.) which were treated in “Abelian Groups” will not be touched upon here. The twin aims of this book are to introduce graduate students to the theory of abelian groups and to provide a young algebraist with a reasonably comprehensive summary of the matenu. >nwhich research in abelian groups can be based. The treatment is by no means intended to be exhaustive or vii

...

Vlll

PREFACE

even to yield a complete record of the present status of the theory-this would have been a Sisyphean task, since the subject has become so extensive and is growing almost from day to day. But the author has tried to be fairly complete in what he considers as the main body of up-to-date abelian group theory, and the reader should get a considerable amount of knowledge of the central ideas, the basic results, and the fundamental methods. To assist the reader in this, numerous exercises accompany the text; some of them are straightforward, others serve as additional theory or contain various complements. The exercises are not used in the text except for other exercises, but the reader is advised to attempt some exercises to get a better understanding of the theory. No mathematical knowledge is presupposed beyond the rudiments of abstract algebra, set theory, and topology; however, a certain maturity in mathematical reasoning is required. The selection of material is unavoidably somewhat subjective. The main emphasis is on structural problems, and proper place is given to homological questions and to some topological considerations. A serious attempt has been made to unify methods, to simplify presentation, and to make the treatment as self-contained as possible. The author has tried to avoid making the discussion too abstract or too technical. With this view in mind, some significant results could not be treated here and maximum generality has not been achieved in those places where this would entail a loss of clarity or a lot of technicalities. Volume I presents what is fundamental in abelian groups together with the homological aspects of the theory, while Volume I1 is devoted to the structure theory and to applications. Each volume has a Bibliography listing those works on abelian groups which are referred to in the text. The author has tried to give credit wherever it belongs. In some instances, however, especially in the exercises, it was nearly impossible to credit ideas to their original discoverers. At the end of each chapter, some comments are made on the topics of the chapter, and some further results and generalizations (also to modules) are mentioned which a reader may wish to pursue. Also, research problems are listed which the author thought interesting. The system of cross-references is self-explanatory. The end of a proof is marked with the symbol 0.Problems which, for some reason or other, seemed to be difficult are often marked by an asterisk, as are some sections which a beginning reader may find it wise to skip. The author is indebted to a number of group theorists for comments and criticisms; sincere thanks are due to all of them. Special thanks go to B. Charles for his numerous helpful comments. The author would like to express his gratitude to the Mathematics Departments of University of Miami, Coral Gables, Florida, and Tulane University, New Orleans, Louisiana, for their assistance in the preparation of the manuscript, and to Academic Press, Inc., for the publication of this book in their prestigious series.

CONTENTS

Preface

vii

I. Preliminaries 1. 2. 3. 4. 5. 6. 7.

Definitions Maps and Diagrams The Most Important Types of Groups Modules Categories of Abelian Groups Functorial Subgroups and Quotient Groups Topologies in Groups Notes

1 6 14 19 21 25 29 34

11. Direct Sums 8. 9. 10. 11, 12. 13.

Direct Sums and Direct Products Direct Summands Pullback and Pushout Diagrams Direct Limits Inverse Limits Completeness and Completions Notes

36 46 51 53 59 65 70

ix

CONTENTS

X

111. Direct Sums of Cyclic Groups 14. 15. 16. 17. 18. 19.

Free Abelian Groups-Defining Relations Finitely Generated Groups Linear Independence and Rank Direct Sums of Cyclic p-Groups Subgroups of Direct Sums of Cyclic Groups Countable Free Groups Notes

72 71 83 87 91 93 95

IV. Divisible Groups 20. Divisibility 21. Injective Groups 22. Systems of Equations 23. The Structure of Divisible Groups 24. The Divisible Hull 25. Finitely Cogenerated Groups Notes

91 99 101 104 106 109 112

V. Pure Subgroups 26. Purity 21. Bounded Pure Subgroups 28. Quotient Groups Modulo Pure Subgroups 29. Pure-Exact Sequences 30. Pure-Projectivity and Pure-Injectivity 31. Generalizations of Purity Notes

113 117 120 122 125 128 133

VI. Basic Subgroups 32. 33. 34. 35. 36. 37.

p-Basic Subgroups Basic Subgroups of p-Groups Further Results on p-Basic Subgroups Different p-Basic Subgroups Basic Subgroups Are Endomorphic Images The Ulm Sequence Notes

135 139 143 147 152 153 158

VII. Algebraically Compact Groups 38. Algebraic Compactness 39. Complete Groups 40. The Structure of Algebraically Compact Groups

159 162 167

CONTENTS

41. Pure-Essential Extensions 42. More about Algebraically Compact Groups Notes

xi 170 174 178

VIII. Homomorphism Groups 43. 44. 45. 46. 47. 48.

Groups of Homomorphisms Exact Sequences for Horn Certain Subgroups of Hom Homomorphism Groups of Torsion Groups Character Groups Duality between Discrete Torsion and 0-Dimensional Compact Groups Notes

180 184 188 193 199 203 207

IX. Groups of Extensions 49. Group Extensions 50. Extensions as Short Exact Sequences 5 I . Exact Sequences for Ext 52. Elementary Properties of Ext 53. The Functor Pext 54. Cotorsion Groups 55. The Structure of Cotorsion Groups 56. The Ulm Factors of Cotorsion Groups 57. Applications to Ext 58. Injective Properties of Cotorsion Groups Notes

209 212 217 22 1 226 232 237 240 243 247 250

X. Tensor and Torsion Products 59. 60. 61. 62. 63.

The Tensor Product Exact Sequences for Tensor Products The Structure of Tensor Products The Torsion Product Exact Sequences for Tor 64.The Structure of Torsion Products Notes

252 257 26 1 264 268 271 214

Bibliography

275

Table of Notations

28 1

Author Index Subject Index

285 2 87

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PRELIMINARIES

The principal purpose of this introductory chapter is to acquaint the reader with the terminology and basic facts of abelian groups which will be used throughout the text. Some of the proofs will be omitted as they are standard and can be found in textbooks on algebra or on group theory. The fundamental types of groups, together with their main properties, are briefly discussed here. We shall save numerous repetitions by the adoption of their conventional notations. Maps, diagrams, categories, and functors are also presented; they will play an important role in o u r developments. Some of the most useful topologies in abelian groups will also be surveyed. A reader not familiar with the subject treated here is advised to read this chapter most carefully.

1.

DEFINITIONS

Abelian groups, like other algebraic systems, are defined on sets. In abelian group theory, however, certain set-theoretical features of the underlying sets seem to play a much more important role than in other parts of algebra. Therefore, we shall frequently have occasion to refer to cardinal and ordinal numbers, and to some results in set theory. In spite of this, we are not going to discuss the set-theoretical backgrounds of abelian groups. We accept the Godel-Bernays axioms of set theory, including the Axionz of Choice which we use mainly in the equivalent form called Zorn’s lemma. Let P be a partially ordered set, i.e., a set with a binary relation 5 such that a 5 a ; a 5 b and b 5 a imply a = b ; a 5 b and b 5 c imply a 2 c, for all a , 6 , c E P. A subset C of P is a chain, if a, b E C implies either a 5 b or b 5 a. The element u E P is an upper bound for C , if c 5 u, for all c E C , 1

I.

2

PRELIMINARIES

and P is said to be inductive, if every chain in P has an upper bound in P. A v E P is maximal in P,if u 5 a with a E P implies a = v.

Zorn’s Lemma. I f a partially ordered set is inductive,then it contains a maxip a l element. Whenever necessary, we assume the Continuum Hypothesis, too; this fact will always be stated explicitly. Class and set will be used as customary in set theory. If we say family or system, then we d o not exclude the repeated use of the same element. We adapt the conventional notations of set theory [see the table of notations, p. 281 ] except for writing LY : U H b to mean that c1 is a function that maps the element a of some class [set] A upon the element b of a class [set] B, while a : A .+ B denotes that c1 is a function mapping the class [set] A into B. The word “group ” will mean, throughout, an additively written abelian [i.e., commutative] group. That is, by group is meant a set A of elements, such that with every pair a, b E A there is associated an element a + b of A , called the sum of a and b ; there is an element O E A , the zero, such that a + 0 = a for every a E A ; to each a E A there exists an x E A with a + x = 0, this x = -a is the inverse of a ; finally, both the associative and the commutative laws hold :

(a

+ b) + c = u + (b + c),

a

+ b = b + a,

for all a, 6, c E A . Note that a group is never empty, because it contains a zero, and that in a n equality a + b = c, any two of a, 6, c uniquely determine the third one. The associative law enables us to write a sum of more than two summands without parentheses, and due to commutativity, the terms of a sum can be permuted. For the sake of brevity, one writes a - b for a + (- b) ; thus -a - b is the inverse of a + b. The sum a + * * . a [n summands] is abbreviated as nu, and -a - . . -a [n summands] as (- n)a or -nu. A sum without terms is 0; accordingly, Oa = 0 for all a E A [notice that we d o not distinguish in notation between the integer 0 and the group element 01. An element na, with n an integer, is called a multiple of a. We shall use the same symbol for a group and for the set of its elements. The order of a group A is the cardinal number IAl of the set of its elements. If IAl is a finite [countable] cardinal, A is called afinite [countable]group. A subset B of A is a subgroup if the elements of B form a group under the same rule of addition. If A is finite, by Lagrange’s theorem, I BI is a divisor of [ A \ .A subgroup of A always contains the zero of A , and a nonempty subset B of A is a subgroup of A if and only if a, b E B implies a + b E B, and a E B implies -a E B, or, more simply if and only if a, b E B implies a - b E B. The trivial subgroups of A are A and the subgroup consisting of 0 alone; there being no danger of confusion, the latter subgroup will also be denoted by 0.

+

3

1. DEFINITIONS

A subgroup of A , different from A , is a proper subgroup of A . We shall write B 5 A [ B < A ] to indicate that B is a subgroup [a proper subgroup] of A . If B 5 A and a E A , the set a + B = { a + b I b E B } is called a coset of A modulo B. Recall that

+

+

(i) b w a b is a one-to-one correspondence between B and a B; (ii) a,, a, E A belong to the same coset mod B if and only if a , - a , E B ; one may write then a , = a2 mod B and say: a,, a , are congruent mod B ; (iii) two cosets are either identical or disjoint; (iv) A is the set-theoretical union of pairwise disjoint cosets of A mod B.

An element of a coset is called a representative of this coset. A set consisting of just one representative from each coset mod B is a complete set of representatives mod B. Its cardinality, i.e., the cardinal number of the set of different cosets mod B, is the index of B in A , denoted as IA : BI. This may be finite or infinite; in the first case, B is offinite index in A . If A is a finite group, then [ A : BI = lAl/lBl. The cosets of A mod B form a group A/B known as the quotient or factor group of A mod B. In A / B , the sum of two elements C,, C2 [which are cosets of A mod B] is defined to be the coset C containing the set {c, + c2 I c1 E C1, c2 E C , } ; actually, this set is itself a coset and thus it is identical with C. The zero element of A / B is B [qua its own coset], and the inverse of a coset C, is the coset - C, = { -c I c E C , } . A / B is a proper quotient group of A if B # 0. We shall frequently refer to the natural one-to-one correspondence between the subgroups of the quotient group A* = A/B and the subgroups of A containing B. The elements of A contained in elements [i.e., cosets of A] of some subgroup C* of A* form a subgroup C such that B 5 C 5 A . On the other hand, if B 5 C 5 A , then the cosets of A mod B containing at least one element from C form a subgroup C* of A * . In this way, C and C* correspond to each other, and we may write C* = C/B. Notice that IC*l = IC: BI, and /A* : C*l = IA : CJ. The set-theoretic intersection B n C of two subgroups B, C of A is again a subgroup of A . More generally, if B, is a family of subgroups of A , then their intersection B = Bi is likewise a subgroup of A . We agree to put B = A if i ranges over the empty set. If S is a subset of A , the symbol ( S ) will denote the subgroup of A generated by S, i.e., the intersection of all subgroups of A containing S. If S consists of the elements a , (i E I ) , we also write

0,

or simply ( S ) = This ( S ) consists of all sums of the form nlal + .. . + nkak [this is called a linear combination of a,, * * . , a,] with a , E S, n, integers, and k a nonnegative integer. If S is empty, then (S) = 0.

I.

4

PRELIMINARIES

If (S) = A, S is said to be a generating system of A ; the elements of S are generators of A. Afinitely generated group is one which has a finite generating system. Notice that (S) is of the same power as S unless S is finite, in which case (S) may be finite or countably infinite. If B and C are subgroups of A , then the subgroup ( B , C ) they generate consists of all elements of A of the form p + c with b E B, c E C. We may write, therefore, ( B , C ) = B + C. For a possibly infinite collection of subgroups B, of A, the subgroup B they generate consists of all finite sums b,, + ... + b,, with b,, belonging to some B,,; we shall then write B = I B, . The group ( a ) is the cyclic group generated by a. The order of ( a ) is also called the order of the element a, in notation: o(a). The order o(a) is thus either a positive integer or the symbol 00. If o(a) = co, all the multiples nu of a (n = 0 , & 1 , k2, . - .) are different and exhaust ( a ) , while if o(a) = m, a positive integer, then 0, a, ..-,(m - l)a are the different elements of ( a ) , and ra = sa if and only if m I r - s. If every element of A is of finite order, A is called a torsion or periodic group, while A is torsionzfree if all its elements, except for 0, are of infinite order. Mixed groups contain both nonzero elements of finite order and elements of infinite order. A primary group or p-group is defined to be a group the orders of whose elements are powers of a fixed prime p .

1,

Theorem 1.1. The set T of all elements of finite order in a group A is a subgroup of A. T is a torsion group and the quotient group AIT is torsion-free. Since 0 E T, T is not empty. If a, b E T, i.e., ma = 0 and nb = 0 for some positive integers m, n, then mn(a - b) = 0, and so a - b E T, T is a subgroup. To show AITtorsion-free, let a + Tbe a coset of finite order, i.e., m(a T) c T for some rn > 0. Then ma E T, and there exists n > 0 with n(ma) = 0. Thus, a is of finite order, a E T, and a + T = T is the zero of A1T.n

+

We shall call T the maximal torsion subgroup or the torsion part of A , and shall denote it by T(A). Note that if B is a torsion subgroup of A, then B 5 T, and if C 5 A such that A / C is torsion-free, then T 5 C . For a group A and an integer n > 0, let nA = {nu I a E A} and A[n]

=

{a I a E A, nu

= O}.

Thus g E nA if and only if the equation n x = g has a solution x in A, and g E A[n] if and only if o(g)I n. Clearly, nA and A[n] are subgroups of A. If a is an element of order p k ,p a prime, we call k the exponent of a, and write e(a) = k. Given a E A, the greatest nonnegative integer r for which p'x =a is solvable for some x E A, is called the p-height h,(a) of a. If p'x = a is solvable whatever r is, a is of infinite p-height, h,(a) = 00. The zero is of

1. DEFINITIONS

5

infinite height at every prime. If it is completely clear from the context which prime p is meant, we call h,(a) simply the height of a and write h(a). The socle S(A) of a group A consists of all a E A such that o(a>is a squarefree integer. S ( A ) is a subgroup of A ; it is 0 if and only if A is torsion-free, and it is equal to A if and only if A is an elementary group in the sense that every element has a square-free order. For ap-group A , we have S(A) = A [ p ] . The set of all subgroups of a group A is partially ordered under the inclusion relation. It is, moreover, a lattice where B n C and B + C are the lattice operations " inf" and " sup " for subgroups B, C of A . This lattice L(A) has a maximum and a minimum element ( A and 0), and it satisfies the modular law: if B, C , D are subgroups of A such that B 5 D , then

B

+ (C n D) = ( B + C) n D.

In fact, the inclusion 5 being evident, we need only prove that every d E ( B + C ) n D belongs to the subgroup on the left member. Write d = b + c with b E B, c E C ; thus d - b = c belongs to D and C . Hence c E C n D, and d = b + c E B + ( C n D),indeed.

EXERCISES 1. Prove that a finite group A contains an element of order p if and only if p divides the order of A . 2. If B < A and IBI < IAI, then IA/BI = ( A ] ,provided IAl is infinite. 3. Let B, C be subgroups of A such that C 5 B and IB : CI is finite. Then, for every subset S of A , ( S , C > is of finite index in ( S , B ) , and this index divides I B : CI. 4. (a) (W. R. Scott) Let Bi(i E I ) be subgroups of A , and let B denote their intersection. Then the index IA : BI is not larger than the product of the [ A :B J ,i E I. (b) The intersection of a finite number of subgroups of finite index is of finite index. 5 . Let B, C be subgroups of A . (a) For every a E A , a + B and a + ( B C ) meet the same cosets mod C. (b) A coset mod B contains I B : ( B n C)l pairwise incongruent elements mod C. 6. (0. Ore) A has a common system of representatives mod two of its subgroups, B and C , if and only if

+

( B : ( B n C ) I= I C : ( B n C ) I . [Hint: for necessity, use Ex. 5; for sufficiency, divide the cosets mod B into blocks mod B + C and make one-to-one correspondences within the blocks.]

6

I.

PRELIMINARIES

7.* (N. H. McCoy) (a) If B, C, G are subgroups of A such that G is contained in the setunion B v C , then either G 5 B or G 5 C. [Hint:if b E ( B n G)\C, then C E C n G implies b + C E B n G, C E B n G.] (b) The same does not hold for the set-union of three subgroups. (c) If G 5 A is contained in the set-union of the subgroups B,, ..., B,, of A , but not in the union of any n - 1 of the B i , then mG 5 B, n * * * n Bn for some integer m > 0. [Hint: apply an argument like the one in (7.3) irzfra.1 Let B 5 A , and let S be a subset of A disjoint from B. There exists a subgroup C of A such that: (i) B 5 C; (ii) C does not intersect S; (iii) C < C’ 5 A implies that C‘ does intersect S. 9. Let B, X be subgroups of A . There exists a subgroup C of A such that: (i) B 5 C ; (ii) B n X = C n X ; (iii) C < C’ < A implies B n X < C‘ n X. 10. (Honda [l]) If B 5 A and m is a positive integer, define 8.

m-l B

=

{a I a E A , ma E B}.

Prove that (a) m-‘B is a subgroup; (b) m-’0 = A rm]; (c) m-’ (mB) = B + A [ m ] ;(d) m(m-’B) = B n m A ; (e) m-’(n-’B) =(mn)-’B. 11. Prove the “triangle inequality for the heights: ”

12.

13. 14. 15.

h,(a + b) 2 min(h,(a), h,(b)), and equality holds if h,(a) # h,(b). If A contains elements of infinite order, then the set of all elements of infinite order in A generates A . If B 5 A , then T(B) = T(A) n B, and S(B) = S(A) n B. For every integer n > 0, T(nA) = nT(A). If B, C, D are subgroups of A , then

+ ( B n D ) S B n (C + D ) B + (C n D ) 5 ( B + C ) n ( B + 0).

(Bn C) and

Find examples where proper inclusions hold. 2. MAPS AND DIAGRAMS

Let A and B be arbitrary groups. A map

c(:A+B [often denoted as A -% B] is a function that associates with each element a E A a unique element b E B, a : a- b. This b is the itnage of a under a, b = a(a), or simply b = cta. A is called the domain and B the range or

2.

7

MAPS AND DIAGRAMS

codomain of a. A map a : A preserves addition, that is, .(al

+B

is a homomorphism [of A into B] if it

+ a 2 )= aa, +@a,

for all a,, a2 E A .

If there is no need to name the homomorphism, we write simply A -+ B. Every homomorphism cx : A + Bgives rise to two subgroups: Ker a 5 A and Im a 5 B. The kernel of a, Ker CI, is the set of all a E A with aa = 0, while the image of a , Im a, consists of all b E B such that some a E A satisfies MU = 6. One may write aA for Im a. If Im cx = B, cx is called surjective or epic; we also say that a is an epimorphism. If Ker cx = 0, a is said to be injective or monic; also, a is a monomorphism. If both Im a = B and Ker a = 0, then a is one-toone between A and B [i.e., it is bijectiw];in this case it is called an isomorphism. The groups A , B are isomorphic [denoted as A E B] if there is an isomorphism a : A + B ; then the inverse map a-1 : B A exists and is again an isomorphism. As customary in algebra, we make no distinction between isomorphic groups, unless they are distinct subgroups of the same larger group considered. If G is a subgroup both of A and B, and if a : A + B fixes the elements of G, then cx is called a homomorphism over G. A homomorphism with 0 image is referred t o as a zero homomorphism; it will be denoted by 0. If A 5 B, then the map that assigns every a E A to itself may be regarded as a homomorphism of A into B ; it is called an injection [or inclusion] map. The injection 0 A is the unique homomorphism of 0 into A . If c( : A + B and C 5 A , then the restriction a I C of a t o C has the domain C and range B, and coincides with a on C. Let a : A + B and P : B + C be homomorphisms; here the range of a is the same as the domain of p. The composite map A -+ B + C , called theproduct of a and p and denoted by fi c( or simply by Pa [notice the order of factors], is again a homomorphism. Recall that Pa acts according to the rule -+

--f

0

(Pcx)a = P(au)

for all a E A .

We have the associative law ?(Pa> = (rPb

whenever the products pa and y/3 are defined. It follows easily that a is rightcancellable [i.e., Pa = ycl always implies fi = y ] exactly if a is an epimorphism, and left-cancellable [aP = ay always implies P = y ] if and only if it is a monomorphism. The product of two epimorphisms [monomorphisms] is again one. A homomorphism of A into itself is called a n endomorphism, a n isomorphism of A with itself an automorphism. The identity automorphism 1, of A satisfies 1,a = a and pl, = p

8

I.

PRELIMINARIES

whenever the left-hand products are defined. A subgroup B of A that is carried into itself by every endomorphism [automorphism] of A is said to be a fully invariant [characteristic] subgroup of A . Let a : A B be an epimorphism, and let Ker a = K . The complete inverse image ci-'b = { a I a E A , cia = b} of an element b E B is a coset a + K in A . It follows that the map a + K ~ c i a[being independent of the special choice of the representative a of the coset] induced by a is an isomorphism between A / K and B. In the same way, every homomorphism ci : A + B induces an isomorphism between A/Ker a and Im a. The mapping a ~ aK is the canonical or natural epimorphism of A onto A / K . If C 5 B 5 A , then 1, induces the epimorphism a + C w a + B of A / C onto A / B . If the homomorphisms a, p have the same domain A and the same range B, then their sum a + p can be defined by the formula --f

+

+ @a = aa + pa for every a E A . It is readily checked that + p : A B is likewise a homomorphism, and one (a

ci

--f

has a+p=p+ci, (a (a

a+o=a,

+ B) + Y = a + (B + Y),

+ p>y = ay + py,

6(a

+ p) = sa + sp,

whenever the sums and products are defined. A sequence of groups A i and homomorphisms ai A,A+A,L+-.*&Ak

(,k2_2)

is exact if Im

cii =

Ker

for i = 1,

a - 1 ,

k

- 1.

B is exact if and only if ci is monic, while In particular, 0 -+ A"B L + C + 0 is exact if and only ifp is epic. The exactness of 0 A"+ B +0 is equivalent to the fact that a is an isomorphism. We call an exact sequence of the form --f

O + A"-

BB-c + o

a short exact sequence; here a is an injection of A into B such that p is an epimorphism with Im ci as kernel. [Notice that in this case A can be identified with the subgroup Im ci of B, and C with the quotient group B / A . ] Roughly speaking, a diagram of groups and homomorphisms consists of capital letters representing groups, and arrows between certain pairs of capital

2.

9

MAPS AND DIAGRAMS

letters representing homomorphisms between the indicated groups. A diagram is commutative if we get the same composite homomorphisms whenever we follow directed arrows along different paths from one group to another group in the diagram. For instance, the diagram

is commutative exactly if the homomorphisms p p and p'a of A into B'coincide, and the same holds for the homomorphisms yv and v'p of B into C'; then the equality of the homomorphisms yvp, v ' p p , v'p'a follows. In diagrams, the identity map will often be denoted by the sign of equality, as, e.g., in

Ip

!I

A Y ' C

This diagram is essentially the same as

If this is commutative, we shall say that y factors through B -+ C. The following two lemmas are rather elementary.

Lemma 2.1. A diagram G

with exact row can be embedded in a commutatioe diagram

G

if and only if pq = 0. Moreorleu, 4 : G + A is unique. If such a q5 exists, then q = a$ implies flq = pa4 = 04 = 0. Thus the stated condition is necessary. Conversely, if pq = 0 in ( l ) , then Im q 5 Ker p. By the exactness of the row, Ker p = Im a,and a is a monomorphism; hence

I.

10

PRELIMINARIES

the map 4 = a-’q of G into A is well defined. It is readily shown to be a homomorphism that makes (2) commutative. If 4’ : G -+ A does the same, then a4’ = q = C Cwhence ~, 4’ = 4, a being a monomorphism.0

In order to save space, diagrams (1) and (2) will be replaced in the future by a single diagram

,G

thus dotted arrows will denote homomorphisms to be

“

filled in.”

Lemma 2.2. A diagram

(3) with exact row can beJilled in by a 4 : C -+ G so as to get a commutative diagram $, and only i f , qa = 0. Moreover, 4 is unique. If such a 4 exists, q = 4j3,then qa = 4j3a = 40 = 0, and the necessity is clear. Conversely, assume qa = 0 and define : C -i G as follows, let 4 c = qb if b E B satisfies /?b = c . This is a good definition, for if b‘ E B also satisfies pb’ = c, then b‘ - b E Ker j3 = Im a 5 Ker q, and so qb‘ = qb. It is readily seen that 4 is a homomorphism making (3) commutative. Finally, if 4‘ : C -+ G also satisfies 4‘p = q, then 4‘j3= 4 j3 and the epimorphic character of j3 imply 4‘ = 4.0 In the proof of the next lemma, we use the procedure of “chasing” elements around diagrams.

Lemma 2.3 (the 5-lemma). Let the diagram A , >-+

A2&

A,&

A,&

lYI 1.; 1.. 1.

A, In

B , A B2 A B381-* B 4 A B ,

be commutative with exact rows. Then (a) i f y l is epic and y 2 , y4 are monic, then y3 is monic; (b) if y s is monic and y 2 , y4 aye epic, then y3 is epic; (c) i f y l is epic, i f y , is monic, and i f y 2 , y4 are isomorphisms, then y3 is an isomorphism.

2.

MAPS AND DIAGRAMS

11

Assume the hypotheses of (a) and let a, E Ker y , . By the commutativity of the third square, y, a, a, = 8, y 3 a, = 0, whence u, a, = 0 because y, is monic. By the exactness of the top row, some a, E A , satisfies u2 a, = a,, and in view of the commutativity of the second square, 8, y2 a, = y 3 a, a, = y , a3 = 0. The bottom row is exact, so PI b, = y, a, for some b, E B,, and y, being epic, yla, =b, for some a , E A,. Thus y 2 ~ , a = l filyla, = P,b, = y,a,, and so alal = a,, for y, is monic. This shows a, = t~, a, = a, alal = 0, i.e., y, is a monomorphism. Next start with the hypotheses of (b), and let b3 E B,. The map y, is epic, so 8, b, = y, a, for some a, E A,. Hence, y s a, a, = 8, y, a, = 8, 8, b3 = 0, and so u4a4 = 0 because y 5 is monic. The top row is exact, therefore a, a, = a4 for some a, E A,, whence 8, y 3 a, = y4u3a, = y4a, = 8, b3 . The exactness of the bottom row and the epic character of y, imply P2 b, = y 3 a, - b, for some b, E B, and y2 a, = b, for some a, E A,. Hence y 3 a, a2 = 8, y, a2 = y, a, - b, ,b, = y3(a, - u2 a,) E Im y 3 , and consequently, y 3 is an epimorphism. Combining (a) and (b), (c) follows readily.0 Another noteworthy lemma of a somewhat different nature is the following, whose proof is again a routine element-chasing.

Lemma 2.4 (the 3 x 3-lemma). Assume that the diagram 0

0

0

is commutatice and all three columns are exact. If thejrst two or the last two rows are exact, then the remaining row is exact. We prove only that the exactness of two first rows implies that the last is exact, while the proof of the other part [that runs dually] will be left to the reader. Assume a3 E Ker u 3 . Since A, is epic, some a, E A , satisfies A, a, = a,. From p, a, a, = u3 1, a, = 0, and the exactness of the middle column, follows the existence of a b, E B, with plb, = ci2a2. From v,j,b, = P z p , b , = 8, ci, a, = 0, we get p,b, = 0, since v, is monic, and so by the exactness of

12

I.

PRELIMINARIES

the first row, some a, E A, satisfies a,a, = 6 , . Hence a2a2 = plbl = plalal = a, I,a,, and so, a, being monic, a, = &a,, whence a , = 2, a, = I , I,al = 0, and a3 is a monomorphism. Since P3 a3I , = P3 p, a, = v2 p2 a, = 0, and I , is epic, we have P3 a, = 0. To show that Ker P3 5 Im a 3 , assume b , E Ker P 3 . We know that some b, E B, satisfies p, b, = b 3 . Thus v , P2 b, = p3 p 2 b, = 0, and the exactness of the third column ensures that some c, E C , satisfies vlcl = P2 b2 . By the exactness of the first row, for some b, E B,, B,b, = c,, whence P2(b2- p l b l ) = p, b, - vlPlbl = 0. From the second row, some a, E A, satisfies a2a2 = b , - p l b l , whence a3I , a, = p 2 a, a, = p, b, = b, , and b3 E Im a s . Finally, Im p3 2 Im P 3 p 2 = Im v 2 P 2 = C3 shows p3 is an epimorphism.0 Let us recall the isomorphism theorems of E. Noether which are used often : (i) If B and C are subgroups of A such that C 5 B, then AIB 2 (A/C)/(B/C),

+

where the natural isomorphism maps a B upon the coset containing a (a E A). (ii) If .B and C are again subgroups of A, then

+C

+ C)/C, being given by 4 : b + (B n C )Hb + C ( b E B).

B/(B n C ) 2 (B the natural isomorphism [This 4 makes the diagram

O+BnC+B-B/(BnC)--+O

commute, the first two vertical maps being inclusion maps.]

EXERCISES L e t a : A - + B a n d P : B + C . Provethat (a) Ker Pa 2 Ker a, and equality holds if fi is a monomorphism; (b) Im pa 5 Im P, and equality.holds if a is an epimorphism. 2. Let again a : A -+ B and /?: B -+ C. (a) If Pa is a monomorphism, then a is monic [but fi need not be a monomorphism]. (b) If Pa is an epimorphism, then j? is epic [but a is not necessarily]. 1.

2.

13

MAPS A N D DIAGRAMS

3. For every homomorphism 0 + Ker or

CI

:A

-+

-+

B there is an exact sequence

A -% B-+ B/Im a -+ 0.

4. If p is the multiplication by the positive integer rn, and if p denotes the inclusion map, then the sequence O-+A[m]~+A-’-+rnA-+O is exact. 5 . Let a : A -+ B and B‘ 5 B. (a) If we write C ’ B ‘ = { a I a E A , ora E B’}, then or(a-’B’) 5 B‘. (b) For A‘ A we have A‘ 5 cr-’(ctA’), where equality does not hold in general. 0 + A L + B B + C -+ 0 is an exact sequence, and B‘ 5 B, then there If 6. exist A’ 5 A and C‘ 5 C such that the sequence 0 -+ A‘ . I + B’ -% C’ -+ 0 is exact where a‘ = a 1 A’ and p’ = p 1 B‘. 7. In a diagram

with exact bottom row, the dotted arrow can be filled in to make the diagram commutative exactly if Im ~ C 5I Im y. Moreover, 4 is unique. 8. Formulate and prove the dual of Ex. 7 [all the arrows are reversed]. 9. There is a homomorphism A A‘ which makes the diagram -+

O-+A-+B-+C-+O

1

0 - + A ’ - + B’ --f C’+O commutative if and only if there is a homomorphism C-+ C’ making it commutative, where both rows are assumed exact. 10. Let a,p be two homomorphisms A B, and assume the existence of a group K and a homomorphism y : K - + A such that ay = by, and if y’ : K ‘ --r A satisfies uy’ = by’, then there is a unique homomorphism 4 : K’ -+ K with 7‘ = 74. Prove that y is a monomorphism and Im y = Ker(or - p). 11 Let B and C be subgroups of A such that A = B + C , and p : B -+ X , y : C X homomorphisms into the same group X . There is an or : A -+ X with a1 B = p and CII C = y if and only if P I B n C = B n C. 12. Prove the second part of (2.4). 13. F o r every positive integer m and for every fully invariant subgroup B of A , the subgroups rnB and B [ m ] are fully invariant in A . --f

-+

14

I.

PRELIMINARIES

14. A fully invariant subgroup of a fully invariant subgroup of A is fully

invariant in A . (a) If B is a fully invariant subgroup of A , and r] is an endomorphism of A , then a + B ~ r ] +a B is an endomorphism of A / B . (b) If B is fully invariant in A , and C / B in A / B , then C is fully invariant in A . 16. (a) If Bi (i E I ) are fully invariant [characteristic] subgroups of A , then so are n B i and B i . (b) Given a subset S of A , there exists a unique minimal fully invariant [characteristic] subgroup of A containing S. This is (. . . , 4S, . . .) with 4 running over all endomorphisms [automorphisms] of A . 17. If B is a fully invariant [characteristic] subgroup of A , and S is a subset of A , such that B n S =@, then there exists a fully invariant [characteristic] subgroup C of A such that: (i) B 5 C, (ii) C n S (iii) if C' is fully invariant [characteristic] in A , and if C < C', then C' n S # 0. 18. Let r] be an endomorphism of A and m an integer >O. Then m-' Ker r] = Ker pr]. 15.

=a,

3. THE MOST IMPORTANT TYPES OF GROUPS

Cyclic groups. They were defined as groups that can be generated by a single element, i.e., they are of the form (a). If A = ( a ) is an infinite cyclic group, then it is isomorphic to the additive group Z of the rational integers 0, f 1, f 2, . . . , an isomorphism being given by the correspondence nu Hn. Thus all infinite cyclic groups are isomorphic:, we denote them by the same symbol Z. Together with a, - a is also a generator for A , but no other nu generates A . A finite cyclic group A = ( a ) of order m consists of the elements 0, a, 2a! ... , (m - 1)a. Because of ma = 0, we compute just as with the integers mod m ; thus A is isomorphic to the additive group Z(m) of residue classes of the rational integers mod m. All finite cyclic groups of the same order m are thus isomorphic; we shall use the notation Z(m) for them. Again, let A = ( a ) be cyclic of finite order m. Along with a, every ka with (k, m) = 1 generates A . In fact, if n > 0 is an integer with n(ka) = 0, then mlnk whence by hypothesis on k , we get ml n. This shows o(ka) = m, and thus ( k a ) = (a). Conversely, if ka generates ( a ) , then o(ka) = m, and if we write (k, m) = d, then md-'ka = kd-'ma = 0 whence o(ka) 5 md-I and so d = 1 . Therefore ( k a ) = ( a ) if, and only if, (k, m) = 1. It follows that Z(m) can be generated by a single element in +(HI) ways ; here 4 is Euler's function. In connection with our notation for cyclic groups, it should always be kept in mind that neither Z nor Z(m) is an abstract group: while it is impossible to distinguish between the two generators of an abstract infinite cyclic group or between the #J(m)generators of an abstract cyclic group of order m, Z and Z(m) have distinguished generators, namely, 1

3.

THE MOST IMPORTANT TYPES OF GROUPS

15

and the residue class of 1 , respectively. [This will turn out to be of importance in certain natural isomorphisms.]

Subgroups of cyclic groups are likeicise cyclic. In order to verify this, let B be a nonzero subgroup of ( a ) , and let n be the smallest positive integer with nu E B. Then all the multiples of nu belong to B, and if sa E B with an integer s = qn + r (0 5 r < n), then ra = sa - q(na) E B implies r = 0, showing that B = ( n u ) . If a is of finite order nz, then n I m. In fact, if u, u are integers such that mu nu = ( m , n), then (m, n)a = mua nva = u(na) E B, and so n 5 ( m , n). For different divisors n (>O) of m, the subgroups ( n u ) are different; thus Z(m) has as many subgroups as m has divisors. Note that, of two subgroups ofZ(m), one contains the other if and only if the corresponding divisor of rn divides the other one. If a is of infinite order, then so is nu (n > 0), and every nonzero subgroup of 2 is an infinite cyclic group. ( n u ) is of index n in ( a ) , and it is the only subgroup of index n. Let A = ( a ) and B = (nu), with n > 0 a divisor of the order of a if this is finite. Then the quotient group A / B may be generated by the coset a B which is evidently of order n ; thus A/BgZ(n). Consequently, all proper quotient groups [epimorphic images] of acyclic group are finite cyclic groups.

+

+

+

Cocyclic groups. A cyclic group can be characterized as a group A con.ining an element a such that any homomorphism 4 : B + A with a E Im $ ta is epic. Dualizing this concept, we shall call a group C cocyclic if there is an element c E C such that 4 : C -+ B and c $ Ker 4 imply that 4 is monic. In this case, c may be called a cogenerator of C. Since every subgroup is a kernel of a homomorphism, a cogenerator c must belong to all nonzero subgroups of C. Hence the intersection of all nonzero subgroups of a cocyclic group C is not zero; this is the smallest subgroup # O of C. Conversely, if a group has a smallest subgroup #O, then the group is cocyclic, and any element #O in the smallest subgroup is a cogenerator. A cyclic group ( a ) of prime power order pk is cocyclic where any element of order p is a cogenerator. This follows from the simple fact that O < ( p k - ' a ) < ... < ( p a ) < ( a ) are the only subgroups of ( a ) . Another type of cocyclic group was discovered by Prufer [l]. Let p denote a prime. The p"th complex roots of unity, with n running over all integers 2 0 , form an infinite multiplicative group; in accordance with our convention, we switch to the additive notation. This group, called a quasicyclic group or a group of type p m [notation: ZCp")], can be defined as follows : it is generated by elements cl, c2 , . . ., c, , . . . , such that pc, = 0 , pc2 = c1, " ' , P C , + , = c,, . - . . (1) Here o(c,) = p", and every element of Z(pm)is a multiple of some c,.

I.

16

PRELIMINARIES

In order to show Z(p") cocyclic, let us choose a proper subgroup B of Z(p"). There is a generator c,+* of a smallest possible index n + 1 which does not belong to B. We claim B = (c,) (if n = 0, B = 0). Clearly c, E B. Furthermore, every b E B may be written in the form b = k c , for some k and m, where k may be assumed not to be divisible by p . If r, s are integers such that k r p"s = 1, then c, = krc, pmsc, = rb E B, and thus m 5 n, b E (c,), establishing B = (c,). Consequently all proper subgroups of Z(p") are finite cyclic groups of order p" (n = 0,I , 2, . . .). These form a chain with respect to inclusion :

+

+

0 < (cl)


= F ( y ? P) F(CY,s>

and

F ( l C > lD>

= IF(C,D)

whenever p,Sp are defined. Letting D E be~ fixed, Ct+F(C, 0 ) and c c ~ F ( c i lo) , give rise to a covariant functor o n % to &, while for a fixed C E %',D H F(C, D),P H F ( I c , 8) yield a contravariant functor on 9 t o 8. Notice that (1) implies that the diagram F(A, 0 ) F(1a.P)

I

F(a. 10) +

F(C, 0)

1

F(1c P)

F(A, B)

F'a'

+

F(C, B )

is commutative. Examples for functors are abundant. The most important ones in abelian groups are those which assign to a group a subgroup o r a quotient group [they are discussed in the next section], and the functors Hom, Ext, 0 , and Tor [defined in Chapters VIII-XI. The following example is of a different type. Let F be the category whose objects are the sequences [ A ] : A , -% AZz2+A3 of groups and homomorphisms subject to the condition x 2 c(, = 0, and whose morphisms are triples [ y l ,y z , y 3 ] of group homomorphisms yi : A i B, making the diagram --f

[ A ] : Al L1+ A2 LZ+ A3 171

4

IYz

.L

1.13

.L

( [ A ] ,[ B ]E 7 )

[ B ] : B1 A+Bz Lz+ B, commute. I t is straightforward to check that .Pis a category. We define the homology firtictor H on 9- to .d as follows. For [ A ] E .7,let H [ A ]= Ker x2/Im m I and let H [ y l ,y z , y 3 ] : H [ A ]+ H [ B ] be the homomorphism

PI ( aeK er I t is evident that: ( 1 ) a E Ker mz implies y z a t Ker pL;(2) a. a' t Ker m 2 and a - a' c Im LX, imply y 2 a - y 2 a' E Im PI; (3) 4 preserves addition. That H satisfies the covariant functor $:a+Im altiyZa-tIm

conditions (i) and (ii) is straightforward to check.

4 2 )

I.

24

PRELIMINARIES

One of the basic questions concerning functors is to find out how they behave for subgroups and quotient groups. This can be investigated conveniently in terms of exact sequences. If F is a covariant functor on d to d [or subcategories of d ] ,and if 0 -+ A -5 B l + C + 0 is an exact sequence, then F is called left or right exact according as

0 +F(A)

>F(B)

'(j)

,F(c) or

-,o

F(A)*+F(B)~+F(c)

is exact; if F is both left and right exact, it is called exact. For a contravariant F , the displayed sequences are replaced by 0 -+ F(C)

F(B)

~

F ( B ) a + F ( A ) and

F(C)

F(B)

>F(B)

F(a)

>F(A)-+ 0 ,

respectively. The subfunctors of the identity [see next section] are always left exact, while quotient functors are right exact. Let F and G be covariant functors on % to 9. By a natural transformation CD : F -+ G is meant a function assigning to each object A E % a morphism + A : F(A) + G(A) in 9 in such a way that for all morphisms ct : A B in %' the diagram (in 9 ) -+

F(A)

+,I

a F(B)

I+.

G ( A ) Go+G(B) commutes. In this case + A is called a natural morphism between F(A) and G ( A ) . The natural character of certain homomorphisms and isomorphisms is of utmost importance. If 4Ais an isomorphism for every A E V , (D is then called a natural equivalence. In the theory of abelian groups, one encounters almost exclusively additive functors, i.e., functors F satisfying

for all ct, P E d whenever ct + p is defined. For an additive functor F on d to d one obtains F(0) = 0, where 0 may stand for the zero group or zero homomorphism. Also, F(ncc) = nF(ct) for every n E Z. Functors on one category to a second category are studied extensively in homological algebra; we refer to Cartan and Eilenberg [I] and MacLane [3].

EXERCISES 1.

Prove that for any ring R, the left R-modules [as objects] and the Rhomomorphisms [as morphisms] form a category.

6.

FUNCTORIAL SUBGROUPS AND QUOTIENT GROUPS

25

2. Prove that the following is a category: the objects are commutative diagrams of the form

3. 4.

5.

6. 7.

with groups A i ,and the morphisms are quadruples (yl, y 2 , y 3 , y4) of group homomorphisms making all squares arising between (2) and another object commutative. A category with one object is essentially a semigroup [of morphisms] with unit element. Call a category V' a subcategory of a category V if: (i) all the objects of V' are objects of V ; (ii) for A , B E V , Map,.(A, B ) is a subset of Map, (A, B ) ; (iii) the product of two morphisms in V' is the same as their composition in V ; (iv) for A E V ' , 1, is the same in V' as in V . Prove that I : A H A, a~ CI (for A, a E V ' ) is a functor on V' to V . Let g1,V 2 be two categories. The product cutegory Vl x V 2 is defined to consist of the objects ( A l , A2) with A i E Vi and morphisms (xl, a 2 ) : (Al, A2) -+ (B,, B,) with a iE g i , where (pl, fi2)(cr1, a 2 ) is defined if and only if plcc1, p 2 a 2 are defined, and then it is equal to (plal, p2 cc2). Prove that this is actually a category. Check that the homology functor as defined above is a functor. The product of natural transformations is again a natural transformation. 6. FUNCTORIAL SUBGROUPS AND QUOTIENT GROUPS

Some of the most important functors in abelian group theory associate with a group A a subgroup or a quotient group of A . Let us discuss briefly this kind of functor. T o begin, we mention a few examples of such functors. The functorial properties are straightforward to verify. Example 1. Let T : d 4 . g be a functor on the category d to the category B of all torsion groups such that, for A E .d,T ( A ) is the torsion part of A , and for a : A + B in d ,T(a)is the restriction map a 1 T ( A ) : T ( A ) T ( B ) . Example 2. If we use the socle S ( A ) of A rather than its torsion part, then we get again a functor S : d + .g[with S(a)= a I S ( A ) ] . Example 3. For a positive integer 11, let the functor M. : d + sd assign to A its subgroup nA, and to a : A + B the induced homomorphism a I nA : nA + nB. Example 4 . Let .d,denote the category of n-hounded groups, i.e., groups G satisfying nG = 0. A functor d + d , is obtained by assigning A [ n ] to A and a 1 A [ n ] to a : A + B.

I.

26

PRELIMINARIES

Example 5. If %? is the category of torsion-free groups, then the function assigning to A EI the quotient group A / T ( A ) and to a : A + B in d the induced homomorphism a* : a T ( A ) H aa T ( B ) [which map is independent of the choice of a in its coset mod T ( A ) ] of A / T ( A ) into B/T(B)is a functor on d to V.

+

+

Example 6. A functor I -+ d , arises if we set A H A/nA for all A E I and U H u* for all a : A + B in I where a* denotes the induced homomorphism a n A H CLU nB.

+

+

In general, assume that we are given a function F that assigns to every group A E d a subgroup F(A) of A , F(A) 5 A , such that if a : A + B is a homomorphism of A into a group B, then crF(A) 5 F(B), i.e., the restriction map tl I F ( A ) sends F(A) into F(B). In this case, if we agree in putting F(U) = a I F ( 4 , then F is a functor s?4 d.We shall call F(A) a functorial subgroup of A . [Notice that a functorial subgroup arises always via a functor F on d to d or to a subcategory of d ;thus it has to be defined for all A E d,even if in a particular case we restrict our attention to a single group A . ] Our examples 1-4 show that T(A), S ( A ) , n A , A [ n ] are functorial subgroups of A . Next assume that F* is a function which lets a quotient group A/A* of A correspond to A , for every A E d,such that if a : A -+ Bis a homomorphism, then a +A * H + ~ B*

is a homomorphism of A/A* into BIB*. In this case, it is easy to verify the functorial properties of F* : d + d.We call F * ( A ) = A / A * a ,functorial quotient group of A . Examples 5 and 6 show that A / T ( A )and AInA are functorial quotient groups of A . There is a close connection between functorial subgroups and quotient groups : Theorem 6.1. F ( A ) is a functorial subgroup of A i f and only i f A / F ( A ) = F*(A) is a functorial quotient group of A . If ct : A 4 B is a homomorphism, then a* : a + A* H cta + B* is a homomorphism A / A * + BIB* exactly if cta E B* for every a E A*. This is equivalent to the condition MA*5 B* stated for functorial subgroups F ( A ) = A*, F(B) = B*.n Let us point out two rather general methods of manufacturing functorial subgroups and functorial quotient groups. In view of the preceding theorem, there is a natural one-to-one correspondence between the classes of functorial subgroups and quotient groups; therefore, we may confine our attention to functorial subgroups only.

6.

27

FUNCTORIAL SUBGROUPS AND QUOTIENT GROUPS

Let X be a class of groups X . With every A ~d we associate two subgroups, namely,

V,(A)

=

",(A)

=

n Ker 4

with $ : A -+ X E X

,$

and

*

with t+b : X

Im $

--f

A ( X E X).

Thus we let $ range over all homomorphisms of A into groups in the class X, and $ over all homomorphisms of groups in X into A . Proposition 6.2. For a j x e d class X, both V, and W , are functors on d to d. Let n : A + B and 4 : B + X E X. Then 4ci is a homomorphism of A into X , and evidently, V,(A) 5 Ker 4ci with 4 running over all B X E X. It follows that ci maps V,(A) into Ker 4 = V,(B), and so V , is a functor d --f d.In order to prove the same for W, , let again ci : A + B, and $ : X -+ A for some X E X. Then a$ : X B , and evidently I m ci$ 5 W,(B). This shows that ",(A) = Im $ is mapped by ci into W,(B).lJ

0

cJI

--f

0,

--f

We illustrate our functors V,, W, by the following examples (a) Let X consist of all cyclic groups of prime order p . Then V,(A) is the Frattini subgroup of A [see Ex. 4 in 31, while W,(A) = S(A),the socle of A . (b) If X consists of all finite cyclic groups, then Vx(A) is the so-called Ulm subgroup of A, which we shall denote by U ( A ) or by A ' . In this case W,(A) is nothing else than T(A). (c) Next, let X contain one group only, namely Z(m). Then V,(A) = mA [this will follow from (17.2)], while Wx(A)= A [ m ] . (d) If X is again a one-element class, X = {Q), then it will result from theorems in Chapter IV that Vx(A) = T(A), while W,(A) = D(A), the maximal divisible subgroup of A .

If F , and F, are functors d A E d , then we write

+d

such that F , ( A ) 5 F,(A) 2 A for every

Fl 5 F ,

5

and call F, a subjicnctor of F , . This relation 5 between functors of the given type defines a partial order in the class 9 of these functors. F has the maximum element E, the identity functor, E(A) = A , and the minimum element 0, the zero functor : O(A) = 0. For obvious reasons, we shall refer to the class 9 as the class of subfunctors of the identity. If F E 9, then the functor F* [as defined in (6.1)] is called a quotienffirnctor of the identity. In F,5 is actually a lattice-order. For if F,, F , E F ,then A H F , ( A ) n F,(A)

and

A w F , ( A ) + F,(A)

give rise to subfunctors of the identity which are inf(F,, F,) and sup(F,, F,). Therefore, we may denote them by F, A F , and F , v F , , respectively.

I.

28

PRELIMINARIES

Moreover, if F i (i E I) is any family of functors in F , then and define their inf and sup, as is readily verified. In addition to lattice-operations, there is a natural way of introducing a multiplication in F :for F , , F , E F,we set ( F I F z ) ( A )= F,(F,(A)), i.e., F , F , is the product ofthe functors F , , F , in the usual sense. Clearly, F , F , E F . Given a subfunctor F of the identity, we define transfinitely the iterated functors F" for ordinals o as follows. Let Fo = E, and let = FF".

If o is a limit ordinal, we define F"

=

A FP.

PO), i.e., { b E A I 6(a, b) < exp( - k + 1)) = a + p k A . Thus (d) follows from (c), while (a) is an obvious consequence of (d).O In a p-group A , the Z-adic and p-adic topologies coincide. In fact, if A is a p-group, and (m,p ) = 1, then mA = A . This is a simple consequence of what has been shown in example 2 in 4. 3. The Prufer topology is defined in terms of the dual ideal D consisting of all U 5 A such that A / U satisfies the minimum condition [see 251. This is always a Hausdorff topology in which all subgroups are closed. 4. In the finite index topology, the subgroups U of finite index of A constitute a base of neighborhoods. This is a Hausdorff topology exactly if the first Ulm subgroup of A vanishes. This is coarser than the Z-adic and the Priifer topologies. Another general method of making a group A into a topological space is to define a set 6 of subgroups S of A closed and to consider all the cosets a +'S ( Q E A )as a subbase of closed sets in A . In this case, the maps

+

XHX+U,

X H

-x

(for every a E A )

are easily seen to be continuous and open. However, addition fails in general to be continuous simultaneously in the variables. Hence A is [a so-called semi-topological, but] in general, not a topological group in this topology which we shall call closed 6-topology. In order to characterize the cases in which A is topological, we first prove a lemma:

Lemma 7.3 (B. H. Neumann). Lct S,, * . . , S, be subgroups of A such that A is the set-theoretic union ofjinitely many cosets (1)

A

= (a,

+ S , ) u . . . u (a, + 5'")

Then one of S,, ' . . , S, is o f j n i t e index in A .

(aiE A ) .

32

I.

PRELIMINARIES

Assume (1) irredundant, in the sense that none of the cosets a, + Si is contained in the union of the others. If all the S , are equal, then they are of index n in A . Assume that among S , , . - ,Sn there are k 2 2 different and that the assertion has been verified for the case of k - 1 different S i . Let S,;-~,S,bedistinctfromS,+, = -.. = Sn (m < n).Byirredundancy,somecoset x S, is not contained in UI=,+l-(ai Si),hencex Sn E (a, + S i ) . But then

+

+

+

rn

m

A = U ( a i + S i ) u U ( a m + ,- x + a i + S i ) u . . . u i=l

i= 1

uy=

m

U(an-x+ai+Si)

i= 1

has only k - 1 distinct subgroups among S,, .* . , S, , so the assertion follows by induction. 0 The following result shows that if a topology on A , defined in terms of a set 6 of closed subgroups, makes A into a topological group, then it is a linear topology defined by the closed subgroups of finite index in 6 [these are open!]. In the following theorem, the topology is not necessarily Hausdorff.

Proposition 7.4 (A. L. S. Corner). Let 6 be a set of subgroups S of the group A and let 6 , consist of all S E 6 withfinite index in A . Then A is a topological group in the closed 6-topology if and only if this is the closed 6,-topology [i.e., the linear topology defined by taking S E 6 , open]. We need only prove the " only if" part. Assume A topological in the closed 6-topology. Given an open set V = A\(a T ) (a E A , T E 6 ) containing 0, there is an open neighborhood U of 0, such that U - U E V. We may write

+

u = A\

u n

i= 1

+ S,)

(Ui

with a, E A , S, E 6, since every open set is the union of open sets of this form. O E u implies O 4 u ~ = l ( a i S,).Since U - U E V implies for every u E U , u - a $ U , therefore u E u:=,(a a, Si), and

+

+ + A = u (a, + Si) u u ( a + a , + Si).

(2)

n

n

i= 1

i= 1

Let S,, S, be of finite and Sm+l,..., Sn of infinite index in A ; by (7.3), 1 m 5 n. Let S = S, n .. n S, which is again of finite index in A . We claim S E V. For otherwise, we choose b E S n (a + T ) , and for i = 1, .. . , m, we have ai + Si = b ai + Si, which is disjoint from S because of 0 E U . Intersecting (2) with S, we find, replacing a by 6 , . a * ,

+

S

u n

=

i=m+

1

[(ai

+ Si) n S] u

u n

i=m+l

[(b

+ a, + Si)

sl.

7.

TOPOLOGIES IN GROUPS

33

Here the nonempty intersections are cosets mod S n Si, hence by (7.3) some S n Sj with m + 1 s j z n is of finite index in S. Then S n S j and hence Sj is of finite index in A ; this contradiction proves S c V . Thus all open sets V of the form A\(a + T ) ( T E6 ) contain a finite intersection S, n . . . n S, with Si E 6, whence the result follows.0 In view of (7.4), we may disregard closed 6-topologies if we adhere to the continuity of the group operation. Following Charles [4], we introduce the concept of functorial topologies. Assume that, for every A E d , there is defined a topology t ( A ) under which A is a topological group. We call t = { [ ( A )I A E d }a functorial topology if every homomorphism in d is continuous. In this sense, the Z-adic, the p-adic, the Priifer, and the finite index topologies are all functorial. A more general method of obtaining a functorial topology is to choose a class X of groups and to take the subgroups Ker 4 with 4 : A + X E X as a subbase of open neighborhoods about 0 in A . This will yield a linear topology on A which will be Hausdorff whenever there are sufficiently many homomorphisms 4, in the sense that for every a E A , a # 0, there are X E X and 4 : A + X with &I # 0. For the orientation of the reader, we now present a result showing that all infinite abelian groups can be equipped with a nondiscrete Hausdorff topology. This result is of theoretical importance, but no use will be made of it.

Theorem 7.5 (KertCsz and Szele [ I ] ) . nondiscrete Hausdorff topological group.

Erwry infinite abelian group can be made into a

In the proof, we need some simple results which we shall prove later on only. Let A be an infinite group. The Prufer topology makes A into a topological group. This topology is discrete exactly if A itself satisfies the minimum condition. Then by (25.1) A contains a subgroup of type p". The embedding of Z ( p " ) in the group of complex numbers z with / z /= 1 induces a nondiscrete Hausdorff topology on Z(p"), and by translations one obtains a nondiscrete topology on A . 0

EXERCISES 1. A [ n ] is closed in any topological group A . 2. (a) Prove that every homomorphism between groups is continuous in the p-adic, Priifer, and finite index topologies. (b) Every epimorphism is an open mapping in the Z-adic, p-adic, Priifer, and finite index topologies. (c) The Z-adic topology of a subgroup of A is finer than the topology induced by the Z-adic topology of A . 3. (a) Prove that, in a group A , the 2-adic and p-adic topologies coincide, if qA = A for every prime q # p. (b) Let p, q be different primes. For which groups are the p-adic and

34

I.

PRELIMINARIES

q-adic topologies the same ? 4. J, is compact in its p-adic topology. 5. Show that every linear topology is a D-topology for some dual ideal D in the lattice of subgroups. 6 . Let B be a subgroup of the topological group A . Prove that: (a) if B is closed, then the natural map A A / B is an open, continuous homomorphism [recall that in A/B, the neighborhoods are the images of those in A ] ; (b) A / B is discrete exactly if B is open; (c) if tl : A -,C is an open and continuous epimorphism, then the topological isomorphism A/Ker tl z C holds. 7. A closed subgroup B of A is nowhere dense [i.e., A\B is dense] if and only if B is not open. 8. Let A have a linear topology. A subgroup B is closed if and only if B is the intersection of open subgroups of A . 9. If B and C are closed subgroups of A , which has a linear topology, then B + C is not necessarily closed. 10. (a) I n the D-topology, B is a dense subgroup of A if and only if B + U = A f ore ve ry U E D . (b)* A subgroup B of A is dense in the 2-adic (or in the finite index) topology ifand only if A / B is divisible [see 201. 11.* A group can be furnished with a nondiscrete linear Hausdorff topology if and only if it does not satisfy the minimum condition. 12.* Every infinite group admits an invariant metric. [Hint: equivalently, there is a nondiscrete topology satisfying the first axiom of countability ; if a is of infinite order, take (2"a) as a base; if it has infinite socle, take an infinite descending chain of type o with 0 intersection as a base; if it contains a Z(p"), argue as in (7.5).] --f

NOTES Commutative groups are called abelian after the Norwegian mathematician Niels Henrik Abel (1802-1829), who studied algebraic equations with commutative Galois groups. Actually, a finite, commutative grouplike structure was considered by C . F. Gauss in 1801, in connection with quadratic forms [he proved a decomposition like (8.4)], but it was only in the last decades of the 19th century when a more or less systematic study of finite abelian groups developed. As the initial restriction of finiteness has been removed from group theory, Levi started the investigations of infinite abelian groups in his Hubilitarionsschri' [l]. Both Levi and, a little later, Prufer (in his epochal papers [l], [2], [3]), restricted themselves to countable groups, but most of the proofs did not really make use of countability. From the 1930's on, abelian groups have received a good deal of attention, especially the contributions made by R. Baer, L. Ya. Kulikov, and T. Szele are significant. In the

NOTES

35

late 1950’s the homological aspects began to play a stimulating role in abelian group theory. Theorem ( 1 . 1 ) is most elementary, but fundamental. In view of it, the structure theory of abelian groups splits into the theories of torsion and torsion-free groups, and investigations of how these are glued together to form mixed groups. It is hard to trace the history of (1.1). It should be noted that it does not generalize to arbitrary modules. If a “torsion,” element a of an R-module M is defined by o ( a ) # 0, then it is not true in general that the torsion elements form a submodule. I t is known, however, that a ring R has the property that, in every R-module M , the toraion elements form a submodule T such that M / T is torsion-free if and only if R is a left Ore domain [i.e., a domain which satisfies the left Ore condition: L1 n L, # 0 for any two nonzero left ideals Ll. L, of R]. For another notion of “torsion” see A. Hattori [Nagoya Math. J . 17, 147-158 (1960)l. S . E. Dickson [Trans. Amer. Math. SOC.121, 223-235 (1966)] gives a systematic treatment of what he calls a “torsion theory” in abelian categories. See also J . M. Maranda [Trans. Amer. Math. SOC. 110, 98-1 35 (1964)] where torsion preradicals are discussed.

Problem 1. List the functorial subgroups in important subcategories of d. Problem 2. Describe the functorial linear topologies for various categories of abelian groups [elementary, bounded, torsion, etc.].

DIRECT SUMS

The concept of direct sum is of utmost importance in the theory of abelian groups. This is due to two facts: first, if a group decomposes into a direct sum, it can be studied by investigating the components in the direct sum, and these are, in several cases, of a simpler structure; secondly, new groups can be constructed as direct sums of known groups. We shall see that almost all the structure theorems on abelian groups involve, explicitly or implicitly, some direct decomposition. There are two ways of introducing direct sums: the internal and external direct sums. Both will be discussed here, along with their basic properties. The external definition leads to unrestricted direct sums, called direct products, which, too, are extremely useful for us. We discuss pullback and pushout diagrams as well. Also, we are going to define direct and inverse limits, which have begun to play an increasing role in the theory of abelian groups. We shall often have occasion to use them in various context. In the final section of this chapter, completions under a linear topology are dealt with. 8.

DIRECT SUMS AND DIRECT PRODUCTS

Let B, C be subgroups of A , and assume that they satisfy (i) B + C = A ; (ii) B n C = 0. In this case we call A the [internal] direct sum of its subgroups B, C, and write

A=BQC. Condition (i) states that every a E A may be written in the form a = b + c (b E B, c E C ) ,and (ii) amounts to the unicity of this form. For, if a = b + c = b' + c' (b' E B, c' E C ) , then b - b' = c' - c E B n C = 0; on the other hand,

36

8.

DIRECT SUMS AND DIRECT PRODUCTS

31

the uniqueness of the form a = b + c excludes the case that b + 0 = 0 + c is a nonzero element common to B and C. If (ii) is satisfied, then we say: the subgroups B and C are disjoint. This is not consistent with the set-theoretical use of this word, but there is no danger of confusion in this abuse. Let Bi ( i E I ) be a family of subgroups of A , subject to the following two conditions:

c

(i) Bi = A [i.e., the Bitogether generate A]: (ii) for every i E I , BincBj=O. j#i

Then A is said to be the direct sum of its subgroups B , , in sign:

A=@B, i E I

or

A=B,@...@B, if I = ( 1 , - - -n,} . Again, every a E A can be written in a unique form a = b,, + . * + b,, with b i j # 0 belonging to different components B,, ( j = 1 , * . . ,k where k 0). Since every element of X B , is contained in a subgroup generated by a finite number of the Bi , condition (ii) can be replaced by the apparently weaker postulate:

.

Bi n ( B i ,

+

* * *

where i j # i, and k is a natural integer. Let a E A = B 0 C, and write u = b

+ Bik)= 0,

+ c with b E B, c E C. The maps

n : a ~ b and

8 : a ~ c

are immediately seen to be epimorphisms of A onto B and C, respectively. Since nb = b, nc = 0, 8c = c, 86 = 0, and nu + 8a = a, the endomorphisms n, 8 of A satisfy:

(1)

n2 = 71,

e2 = 0,

8n=718=0,

7 1 + 8 = l A .

If we mean by a projection an idempo tent endomorphism and by orthogonal endomorphisms those with 0 products [in both orders], then (1) may be expressed by saying that a direct decomposition A = B Q C dejnes a pair of orthogonal projections with sum 1,. Conversely, any pair n, 6 of endomorphisms of A satisfying ( 1 ) yields a direct decomposition A = nA@BA.

38

11.

DIRECT SUMS

In fact, idempotency and orthogonality imply that an element common to nA and BA must be both reproduced and annihilated by n and 6, so nA n 8A = 0, while n + 6 = 1, guarantees that nA + OA = A . Even if A is a direct sum of several subgroups, A = @ Bi , the decomposition can be described in terms of pairwise orthogonal projections. The ith projection n i: A -,Bi assigns to a = b,, + . . . + b,, the term bi from Bi [which may very well be 01. Then we have: (a) n i nj = 0 or x i , according as i # j or i = j ; (b) for every a E A , almost all of nia are 0, and their sum equals a. Conversely, if ni (i E I ) is a set of endomorphisms of A satisfying (a) and (b), then A is the direct sum of the ni A , as is readily verified. A subgroup B of A is called a direct summand of A , if there is a C 5 A such that A = B 0 C . In this case, C is a complementary direct summand, or simply a complement of B in A . Some of the most useful properties of direct sums are listed as follows: (a) If A = B 0 C, then C 2 A / B . Thus the complement of B in A is unique up to isomorphism. (b) If A = B @ C, and if G is a subgroup of A containing B, then G=B@(GnC). (c) If aE A = B @ C, and if a = b + c (b E B, C E C), then o(a) is theleast common multiple of o(b) and o(c). (d) If A = @ Bi , and if, for every i, C i 5 B, , then C , = C , . This is a proper subgroup of A if Ci< B, for at least one i. (e) If A = @ B , , where each B , is a direct sum, Bi = @ B i j, then

,

1

A= @ @ i

j

Bij.

The latter is called a reJinement of the first decomposition of A . (f) If A = O i B j B i j ,then A = O i B i with B , = O j B i j . Two direct decompositions of A , A = @ Bi and A = @ C j , are called isomorphic, if one can find a one-to-one correspondence between the two sets of components B , and C j such that corresponding components are isomorphic. It is convenient to call an exact sequence o B AA L c + o splitting if Im CI is a direct summand of A . --f

Now we have come to the definition of the external direct sum. Given the groups B and C, we wish to have a group A that is the direct sum of two of its subgroups, B' and C', such that B' E B, C' E C . The set of all pairs (b, c) with b E B, c E C forms a group A under the rules: if and only if b, = b, , c1 = c 2 , (b1, ~ 1 + ) (bz cz) = (b1 b, ~1 + ~ 2 ) .

(bl, cl) = (b, , c,)

3

+

9

8.

DIRECT SUMS AND DIRECT PRODUCTS

39

The correspondences b H (b, 0) and c H (0, c) are isomorphisms of B, C with subgroups B', C' of A . We have A = B' @ C'. If we think of B, C being identified with B', C' under the mentioned isomorphisms, then we may write A = B 0 C and call A the [external] direct sum of B and C. Let Bi (i E I ) be a set of groups. A uector (. . , b i ,- ..) over this set of Bi has exactly one coordinate bi for each i E Z, and bi E Bi. Such a vector may also be interpreted as a function f defined over Z such that f ( i ) = bi E Bifor every i E I. Equality and addition of vectors are defined coordinatewise [i.e., for functions, pointwise]. In this way, the set of all vectors becomes a group C called the direct product [Cartesian sum or complete direct sum] of the groups

Bi: C=

nB i .

isl

The correspondence pi : bi H(. .. ,0, bi, 0, * * *),

where bi stands on the ith place and 0 everywhere else, is an isomorphism of

Biwith a subgroup Bfof C. It is easy to see that the groups Bf (i E I ) generate in C the group A of all vectors (. . * , b i, . . .) with bi = 0 for almost all i E I,

and A is just their direct sum. A is also called the [external] direct sum of the Bi , A = @ Bi. Clearly, A = C if and only if Z is a finite set. We leave it to the reader to show that both the direct sum and the direct product are, up to isomorphism, uniquely determined by the set of components Bi. For a group A and for a cardinal 111, @ , , A will denote the direct sum of m copies of A and the symbol A = A" the direct product of lit copies of A . In a similar fashion, A' stands for the direct product of groups isomorphic to A and indexed by I . Clearly, A' is the group of all functions on Z to A , and we have A' z A'''. The external direct sums and direct products can also be described in terms of systems of homomorphisms. The functions

n,,,

(b, o), nB : (b, c) H b,

p B

:b

(0, c), nc : (6, c) H c,

pC

:c

are homomorphisms as indicated by the diagram

B c B O C - C , Pa

PC

RE

nc

called [coordinate] injections and projections, respectively. They satisfy ~ B P B = ~ B ,

ncpc=lc,

~ B P , = ~ C P B = O ,

~BnB+~cnc=IBec.

11.

40

DIRECT SUMS

More generally, the direct sum [product] of the Bi ( i E I ) defines for every i E I, an injection p i and a projection ni

Bi pi+ 0 Bi+’

n

Bi

[B, L+ Bi 2 + B i ] wherepibi = ( . . . , O , bi,O, ...),ni(...,bj;..,bi;..) = bisatisfythefollowing conditions : (a) ni p = 1B i or 0, according as i = j or i # j ; (b)if I is a finite set, Z = { l ; . - , n } , and B = B , @ . * . @ B , , , then p1n1+ * * . pn7Tn= 1,.

+

If, however, I is infinite, then for direct sum replace (b) by: (b’) each b E @ Bi can be written as a finite sum b = pi,nilb

+ ..-

+ Pinninb, and for direct product by: (b”) given a set {bi} with exactly one bi E Bi for each i E I , there exists a unique b E Bi such that x i b = bi for every i.

n

The following two results exhibit useful universal properties of direct sums and direct products.

Theorem 8.1. Let

+i:Bi + A be homomorphisms, i E I. In the diagrams

with pi the injection maps, the dotted arrow can befilled in uniquely by some $ [independently of i ] to make all diagrams commutative. Write b E @ Bi in the form b = pln,b . + P k f l k b [where 7ci are the corresponding projections], and define $b = c$,n,b + . . . + nkb. This $ is easily seen to be a homomorphism into A satisfying $pi bi = c$i 7ci pi bi = $ibi . This b,t must be unique, for if $’ : @ Bi + A also makes (2) commutative, then I ! , I - $’ vanishes on all p i bi, hence on all b E @ B i , that is, $ - $’ = 0 . 0

+

Theorem 8.2. Let c$i: A + B , be homomorphisms, i unique homomorphism $ making all the diagrams

commute; here x i denote the projections.

E

I. There exists a

8.

41

DIRECT SUMS AND DIRECT PRODUCTS

I7

For a E A , define $a as the unique b E Bi with n i b = 4 i a [cf. (b”)]. This $ is evidently a homomorphism such that r i $ = 4 i for every i. It is unique, for if $’ is a homomorphism with the same stated property, then xi($ - $’)a = 0 for every i, and thus ($ - $’)a E Bi has only 0 coordinates. That is, ($ - $’)a = 0 for every a E A , $ -’)I = 0 . 0

n

Let us introduce the following notations. If A , and Bi (i E I ) are groups, and a i : A i + B , are homomorphisms between them, then there exist unique homomorphisms a : A = @ A i - + B = @ B, and a * : A* = n A i + B * = Bi I

fl

i

such that n!ap. , , = a.,= n!a*p. I ,

and

njapi=O =nJa*p,

(i#j)

for the injections p i of the A i and projections n: of the B i . Namely, a [a*] sends the ith coordinate a, upon the ith coordinate a i a i . We shall denote them as a =@ai and a* = n a i . i

i

For a group G, we introduce two maps: the diagonal map AG : G [the number of components can be arbitrary] as A,: g - ( . . . , g , ..., g ,

...)

-+

n

G

( 9 E GI,

,

and the codiagonal map V , : 0 G + G as V G : ( . . * , g i , * * * ) ~ C (ggii E G ) . i

If there is no danger of confusion, we may suppress the index G. Lemma 8.3. If the diagram of (2.4) has exact rows and columns, and is commutative, then the following sequences are exact: O+ A , .211+ B , Wz@Pz)A ’B 3 @ c2 and V(UZf€JPl) A, OBI ’B2 v z P 2 * C 3 + 0 . 3

Since A, and a, are monic, so is a, Al. Furthermore, (p, 0 P2)Aa2A1 = 0, because /?, a, = O and p, a, 1, = p , p1a1= 0. If b, E B, belongs to Ker(p, @ P2)A, then both p 2 b 2= 0 and P2b2= O . We can find a b, E B, such that p, h, = b,, and here b, E Im a,, since P1b, = 0 follows from vlPlb, = P 2 p l b l = P, b, = 0. Therefore b , E Im p , a l = Im a , A,. The exactness of the second sequence follows by a similar argument. 0 Fora=(..-,bi;.-)EflBidefine

thesupport o f a a s

s(a) = {i E I 1 bi #

O}.

42

11.

DIRECT SUMS

If K is an ideal of the Boolean algebra B of all subsets of 1, then by the K-direct sum of the Bi we mean the subset of Bi whose elements are the vectors a with s(a) E K. Since s(al - a z ) s s(aJ u s(a,), the K-direct sum is a subgroup of B i ; it will be denoted by the symbol

n

n

OKBi.

If, in particular, K consists of all finite subsets of 1, then we get the direct sum, while if K = B, then we arrive at the direct product. Among the subgroups of the direct product there is an important type which frequently occurs in algebra. A subgroup G of the direct product A = Bi is said to be a subdirect sum of the Bi if, for every i, the map x iI G : G -+ B , is an epimorphism. This means, that for each i E Z and for every bi E Bi , G contains a vector (. . . , b i, . . .) whose ith coordinate is just bi . In this case, Bi is not necessarily a subgroup of G, it is an epimorphic image of G under ni I G = q i . Clearly,

n

Ker q i = 0. i

Conversely, if K i(i E I ) is a family of subgroups of G such that OKi = 0, then G is a subdirect sum of the quotient groups G / K i under the monomorphism gH(.-,g

+K~,.-)E~~(G/K~). i

There are a great number of subdirect sums in a direct product of groups, and no complete survey of subdirect sums is known except for the case of subdirect sums of two groups. Assume that the groups B, C are mapped by epimorphisms /?,y upon a group F. The elements (6, c) E B @ C with /?b= yc form a subgroup A of B @ C , and it is routine to check that A is a subdirect sum of B and C. Conversely, if A is a subdirect sum of B and C, then the elements b E B with (6, 0) E A form a subgroup B, 5 B, and the elements c E C with (0, c) E A form a subgroup C, 5 C. It is straightforward to verify that the correspondence b

+ Bo H C + Co

whenever (6, c) E A is an isomorphism of BIB, with C / C , . Now A consists exactly of the pairs (b, c) (b E B, c E C) for which the canonical epimorphisms B --+ BIB,, C -+ CjC, map b and c upon corresponding cosets. This shows that all subdirect sums of two groups B, C arise in the manner described at the beginning of this paragraph. The groups B , , C , are called the kernels of the subdirect sum ; the subdirect sum is in general not determined by the kernels, it does depend also on the choice of /?,y . If B,, C, are considered as subgroups of A , then the following isomorphisms hold:

8.

43

DIRECT SUMS AND DIRECT PRODUCTS

A/(Bo 0 C,)

s'

BIB,

s'

C/C,,

A/B,

s'

C,

AIC, z B.

One of the most important applications of direct sum is the next theorem. Theorem 8.4. A torsion group A is the direct sum of p-groups A , belonging to different primes p. The A , are uniquely determined by A . Let A , consist of all a E A whose order is a power of the prime p. In view of 0 E A , , A , is not empty. If a, b E A , , that is, if p"a = p"b = 0 for some integers m, n 2 0, then p""" (m")(a- b) = 0, a - b E A , , and A , is a subgroup. Every element in A,, .* * A,, is annihilated by a product of powers of pl, .. . , pk, therefore

+ +

A , n (A,,

+ . . - + APk)= 0

whenever p # pl, . . . ,pk. Thus the A , generate their direct sum 0,A , in A . In order to show that every a E A lies in this direct sum, let o(a) = m = p;' . p: with different primes p i . The numbers mi = mp;" (i = 1, n) are relatively prime, hence there are integers sl, . , s, such that slml * * s,, m, = 1. Thus a = slmla .. + s, m, a where mi a E A,, [in view of pi' mi a = ma = 01, and so a E A,, + . . . + A," I 0, A,. If A = 0, B , is any direct decomposition of A into p-groups B, with different primes p, then by the definition of the A , , we have B, 6 A , for every p . Since the B, and the A , generate direct sums which are both equal to A , we must have B, = A , for every p . 0 a * * ,

+

+

+ -

The subgroups A , are called the p-components of A . They are, as seen from their definition, fully invariant subgroups of A . If A is not torsion, the p-component of its torsion part T(A) can be called the p-component of A [even if it fails to be a direct summand of A ] . By virtue of (8.4), the theory of torsion groups is essentially reduced to that of primary groups. Example 1. Let rn =p;' . . . p F be the canonical form of the integer m > 0. The pcomponents of Z(m) [as subgroups of a cyclic group] are again cyclic, and the product of their orders is just m. Consequently, Z(m) = Z(p?) @ . . ' @ Z(p2). Example 2. The group Q/Z of all complex roots of unity is a torsion group whose p-component consists evidently of all p"th roots of unity (n = 1,2, . . .). This means the p-component of Q/Z is Z(p"). Hence

Q/Z = 0Z(P"). P

We conclude this section with:

Theorem 8.5. An elementary p-group is the direct sum of cyclic groups of order p .

44

11.

DIRECT SUMS

We show that an elementary p-group A is in the natural way a vector space over the field F, of p elements. In fact, p a = 0 for a E A , and so for n, m E Z,one has na = ma exactly if n = m mod p , that is, n, m represent the same element of F, . It is now routine to check the vector space axioms. Therefore, A as a vector space over a field F, has a basis, say, { a i } i s l .It follows that A = @ isr(ai).O Example 3. Let m be an infinite cardinal number. Then Z(P)" = 0Z(P). 2m

In fact, Z(p)m is by (8.5) a direct sum of groups Z(p), and its cardinality is obviously pm = 2m.

EXERCISES 1. If m=p',' **-PI;",then AImA is the direct sum of the groups A/pyA (i = 1, * * . , k). 2. (a) If A is the direct sum [product] of the groups Bi (i E I ) , and Ci are subgroups of Bi, and if C is the direct sum [product] of the C i , then C is a subgroup of A such that A/C is the direct sum [product] of the quotient groups Bi/Ci. (b) If 0 -+ Ai* Bi* Ci -+ 0 are exact sequences (i E I ) , then so are O + @Ai 3 @ B i

o+n~

and

881

@Ci+O

~B 3, ci-+o. A n~

3. A is a direct sum of its subgroups A i (i E I ) if and only if

CAi= A i

and

n A: i

=O

where A:=CAj. j+i

(a) The direct sum of p-groups [torsion groups] is again a p-group [torsion group]. (b) Determine when the direct product of torsion groups is again a torsion group. 5. (a) If G is a subgroup of A = B @ C, then G is a subdirect sum of B n (G C) and (B G) n C. (b) If G is a subdirect sum of B and C, then B + G = B @ C = G + C. 6. Let A = B @ C = B @ ( @ A i ) . Then B @ C i = B @ A i and C = O i C i hold for Ci = (B + A i ) n C . 7. (a) Let G be a subgroup of the direct sum B 0 C. There exist subgroups B,, B, of Band C , , C , of C such that B, 5 B,, C , 5 C,, Bl @ C, is the minimal direct sum containing G, and B, @ C , is the maximal direct sum contained in G, with components in B, C.

4.

+

+

8.

DIRECT SUMS AND DIRECT PRODUCTS

45

(b) Establish the isomorphisms 4IB2

= G/C,

(B, 0 C,)/G

= G/(B, 0 C,). +

8. (E. A. Walker [l]) (a) Let A = B 0 C = B* @ C*. Then B B* = B @ C , = B*@CT where C, = C n ( B + B*), C: = C* n ( B + B*). (b) A = B 0 C = B* @ C* implies B/(B n B*) E C,* and B*/(B n B*)

=

c,.

9. (F. Loonstra) Call a subdirect sum G of groups A i (iE I ) special, if there exist a group F and epimorphisms ai : A i -+ F (i E I ) such that G consists of all (.. . , a i , .. . , a j , - ..) E A i with . .. = a I. a1. = . . *

fl

=a.a.= J

J

(a) Show that for 111 2 3, not every subdirect sum is special. Ker a i ) H (b) If Z is finite, G is special if, and only if, the maps g + (0 xjg Ker a j [where nj : G -+ A j is the coordinate projection] are all isomorphisms between G/@Ker ai and Aj/Ker a j , for everyj E I. (a) A subdirect sum of Z(p") and Z(p") [for integers n 2 m > 01 with and Z ( P " - ~is) isomorphic to Z(p")@ Z(P*-~). kernels Z(pmVk) (b) A subdirect sum of Z(p") and Z(p") with the kernels Z(p") and Z(p") (n 2 m) is isomorphic to Z(p") 0 Z(p"). (c) Every subdirect sum of a cyclic and a quasicyclic group is direct. There are nonisomorphic groups among the subdirect sums of the groups B = Z(p2)@ Z(p4) and C = Z ( p 3 )0 Z(p') with kernels B, = Z(P)0 Z ( p 3 )and Co = Z ( p 2 )0 Z(P"). Call A subdirectly irreducible if in any representation of A as a subdirect sum of groups A i , one of the coordinate projections A + A i is an isomorphism. Show that A is subdirectly irreducible if and only if it is cocyclic. Let Bi (i E I ) be torsion subgroups of A . The Bi generate their direct sum if and only if the socles S(Bi)generate their own direct sum. Let B, C be subgroups of A , and let B @ C denote their external direct sum. Prove the existence of an exact sequence

+

10.

11.

12.

13. 14.

O + B n C-+ B @ C + B + C + O . 15.

Assume that A has two sequences of subgroups, B,, . c,, - * . , C,,, * * * such that C, = A ,

C,, = B,,@ C,,,

a ,

B,, ,

*

- and

(n = 1,2, ...)

and

nc,, = 0. n

Prove that A is a subdirect sum of the B,, which contains their direct sum.

46

11.

DIRECT SUMS

16. Let G be a subgroup of the direct product A = n A i such that, along with g, all the elements of A with the same support belong to G . If all the A i are of order 2 3 , then G is a K-direct sum. 17. (Dlab 121) Prove that the Frattini subgroup of a direct sum is the direct sum of the Frattini subgroups. 18. If F is the Frattini subgroup of a p-group A, then A / F is the direct sum of cyclic groups of order p. [This does not generalize to arbitrary groups; A = Z ( p ) is a counterexample.]

n,

9. DIRECT SUMMANDS

We have called a subgroup B of A a direct summand of A if A = B @C for some C 5 A. For the projections IC : A -+ B, 0 : A C conditions (1) in 8 hold. Next we focus our attention on B. [It should be emphasized that B alone, in general, does not define 71 uniquely; but it does if C is also known.] We then have the following useful lemma. --f

Lemma 9.1. If there is a projection IC of A onto its subgroup B, then B is a direct summand of A . The map B = 1, - 71 is an endomorphism of A , satisfying conditions (1) in 8. Hence we have A = B @ BA. Here 8A is nothing else than the kernel of 71.0 A rather trivial criterion for B ( 5A) to be a direct summand of A is that the cosets of A mod B have representatives which form a subgroup C ( S A ) . Namely, we then have B + C = A and B n C = 0, thus A = B @ C. Less trivial criteria are formulated in the next result.

Proposition 9.2. For a subgroup B of A the following conditions are equivalent [p : B A denotes the injection map]: (a) B is a direct summand of A ; (b) there exists a commutative diagram -+

9.

47

DIRECT SUMMANDS

is a commutative diagram with exact rows, then there is a homomorphism 4 : V --f B such that the upper triangle is commutative. Assuming (a), we first prove (c). If the hypotheses of (c) hold, and if x : A B is a projection, then 4 = nci satisfies +y = nay = xpj3 = j3. Next assume (c), and choose U = B, V = A , y = p, ci = 1, and p = 1,. Then (b) follows at once. Finally, if (b) holds, then n is a projection of A onto B and hence (9.1) yields (a).O --f

Notice that (b) amounts to saying that the identity automorphism of B can be extended to an endomorphism of A into B. For the sake of future reference we mention the following lemmas.

Lemma 9.3. I f A = B 0 C and if G is a fully invariant subgroup of A , then

G = (G n B) 0(G n C ) . Let n, 0 be the projections attached to the direct decomposition A = B 0 C. By the full invariance of G, nG and OG are subgroups of G. Now nC 5 B and OG 5 C imply zcG n OG = 0, while g = ng + Og ( g E G) implies G = nG + QG, so that G = xG 0 OG. Obviously, nG 5 G n B and OG 5 G n C, and so necessarily nG = G n B and OG = G n C.0

Lemma 9.4 (Kaplansky [I]). I f the quotient group AIB is a direct sum, AIB

=

0 (AiIB), i

and if B is a direct summand of ecery A i , say A i = B @ C i , then B is a direct summand of A , A = B @ ( @ Ci). i

It is clear that B and the Ci generate A . Assume we have b + c1 , n). Passing mod B, we obtain (c, B ) * . . (c, B ) = B. Since c j B E A j / B ,therefore c1 B = ...-- c, B = B . Thus c j E B for every j , and so c j E B n Cj = 0, finally, b = 0. Consequently, B and the C i generate their direct sum.0

+ .. . + c,, = 0 for some b E B and c j E C j ( j = 1, ..

+

+ + + +

+

+

If B is a direct summand of A , then the complement of B in A is unique up to isomorphism, but it is far from being unique as a subgroup. The following result tells us that from one complement all the others can easily be obtained.

Lemma 9.5. Let A

=B

0 C be a direct decomposition with projections

zc, 0. I f A = B O C, with projections zcl, O , , then

(1)

zcl =

+ .4e,

8, = o -

n+e

for some endomorphism 4 of A . Conversely, if zc,, O1 are of the form (1) with some endomorphism of A , then A = B 00,A.

11.

48

DIRECT SUMS

If n,, 0, are attached to A = B 0 C,, then set 4 = 0 - 0,. Clearly, B 2 Ker 4, thus 4 = 4n 40 = 46. If a = b + c = b, + c1 with b, b, E B, c E C, c1 E C,, then 4a = c - c1 = b, - b E B whence n4 = 4. Therefore 0, = 0 - 4 = 0 - n40 and n, = 1, - 0, = n + 0 - 0, = n + 7140. Conversely, if z,, are of the form (l), then n, + 0, = I,, n: = n,, 6: = 0, and n16, = 0,n1 = 0, thus A = n,A @ 0,A. Here Im n1 5 Im n and nlB = nB = B, whence n,A = B . 0

+

Theorem 9.6 (Gratzer and Schmidt [l]). If B is a direct summand of A , then the intersection of all complements C of B in A is the maximal fully invariant subgroup of A that is disjoint from B. Let A = B 0 C with projections n, 6, and let 4 be an endomorphism of A . By (9.3, C, = (0 - n40)A is again a complement of B, therefore for c E we have (0 - n46)c = c and 6c = c [for both ' c E C, and c E C], whence z 4 c = 0. This shows 4 c E C, and since C was an arbitrary complement of B, we obtain 4 c E c, that is, is fully invariant in A . Evidently, B n = 0. If X is any fully invariant subgroup of A with B n X = 0, then by virtue of (9.3),X=(XnB)@(XnC)=XnC;thusX2CandX5cC".0

c

e

e

c

Corollary 9.7. A complement of a direct summand of A is unique fi and only if it is fully invariant in A . 0 If a subgroup B of a group A is to be shown a direct summand of A , then in most cases it is not possible to find directly a projection A B. One then tries to find a complement C to B among the subgroups G of A satisfying GnB=0. Call a subgroup H of A a B-high subgroup (Irwin and Walker [l]) if --f

H nB

= 0,

and if

H < H' 5 A

implies H' n B # 0.

That is, H is maximal with respect to the property of being disjoint from B. Then, in particular, H + B = H O B . The existence of B-high subgroups, for every B, is guaranteed by Zorn's lemma. Moreover, H may be chosen so as to contain a prescribed subgroup G of A with G n B = 0. In fact, then the set of all subgroups of A that contain G and are disjoint from B is not empty and is inductive; thus it contains a maximal member H. A complement C of B in A is evidently a B-high subgroup; moreover, in view of (9.6), it must contain every fully invariant subgroup X of A with X n B = 0. In several cases, the construction of a complement to B consists in selecting a B-high subgroup containing such an X [cf., e.g., the proof of (27. I)]. Let us turn to the proofs of two technical lemmas.

Lemma 9.8. If B is a subgroup of A and C is a B-high subgroup of A , then A.

a E A , p a E C [ p aprime] implies a E B @ C

9.

49

DIRECT SUMMANDS

If a E C , there is nothing to be proved. If a 4 C , then (C, a ) contains, owing to the choice of C, an element b E B, b # 0, i.e., b = c ka for some c E C and integer k . Here ( k , p ) = 1 because of pa E C and B n C = 0. Thus, for some integers r, s, rk + sp = I , and so a = r(ka) + s(pa) = r(b - c) +s(pa)~BOC.0

+

Lemma 9.9 (G. Gratzer). Let A , B, C be as in (9.8). Then A = B 0C ifand only g p a = b c (a E A , b E B, c E C ) implies pb' = b for some b' E B. If A = B @ C and a = b ' + c ' ( b ' E B , c ' E C ) , then p a = p b ' + p c ' = b + c implies pb' = b. Conversely, if pa = b + c implies pb' = b for some b' E B, then a - b' satisfies the hypotheses of the preceding lemma, and so a - 6' E B 0C , a E B @ C. This shows that the quotient group A/(B @ C) contains no elements of prime order, and therefore it is torsion-free. But if x E A is arbitrary, not in B 0C, then ( C , x ) intersects B in a nonzero element b", c" Ix = b" for some c" E C and integer 1. I # 0 because of B n C = 0, thus Ix = b" - C" E B 0C, and A / ( B 0C ) is a torsion group. We conclude A = B 0 C . 0

+

+

A subgroup G of A is called an absolute direct summand of A if, for every G-high subgroup H of A , one has A = G 0H. Absolute direct summands are rare phenomena; they are described in Ex. 8. We conclude the discussion of direct summands with the following result.

Proposition 9.10 (Kaplansky [3], C . P. Walker [I]). Let A be a direct sum of groups A i (i E I ) such that euery A iis at most of power m where in is afixed infinite cardinal. Then any direct summand of A is again a direct sum of groups of power g m . Let A = B 0 C, and consider some summand A , . If { a k } k is a generating system of A , ;and if a, = bk ck (bkE B, c k E C), then every b, and every ck has but a finite number of nonzero coordinates in the direct sum A = @ A i . Therefore, if we collect all the A containing at least one nonzero coordinate of some b, or ck (k E K ) , and take their union in A , we obviously get a direct summand AS of A , of cardinality s i n . Selecting a generating system for A J and repeating the same process with A j replaced by A ; , we find a direct summand A; of A again of power 5 in, etc. The union of the ascending chain A , 5 A J 5 A ; 5 * . . is a direct summand A: of A , such that

+

lAT1 5 tit

and

AT = ( B n AT) 0(C n AT).

A well-ordered, increasing sequence of subgroups S, of A is now defined as follows. Put So = 0. If S, is defined for some ordinal and S,, # A , then pick out some A , not in S,, and let So+,= S, + A:. If CT is a limit ordinal, then we set S, = S, . It is evident that for some ordinal T, not exceeding J A J ,the equality S, = A will hold. It is also clear that So+,/Sn is of power

up= i. Since p i 4 , = $ j n { and 4 j is a monomorphism, .{ai = 0, whence r i a i = 0 and a = 0. This shows 4 , monic.0 The following interpretation of ( I 1.2) is of interest. Given a directed set I , consider the category 9,of all direct systems of abelian groups, indexed by I , with homomorphisms as defined above. Then the function ,!.*assigning to a direct system A its limit A, and to a homomorphism C#J : A + B the homomorphism 4, : A , + B , is, by virtueof ( I 1.2), a functor on 9,to d.In view of this, we are justified to consider 4, as a natural map between direct limits.

Finally, we focus our attention on three direct systems: A , B as above and C = {Ci ( i E I ) ; r~:.}, all with the same index set I . If 4 : A + B and II/ : B + C are homomorphisms between them such that, for every i, the sequence

is exact, then we say, the sequence

11.

58

DIRECT SUMS

is exact. We have: Theorem 11.3. Zf A , B , C are direct systems of groups such that (6) is an exact sequence, then the sequence A,

=

LIJ A i >+4

B,

= & Bi +'

* c, = LIJ c;

under the induced maps 4 * , $, [cf.preceding theorem] is likewise exact. By (1 1.2) the diagram A . Qi+B. A+ ci

.il'

A,-%

(i EI)

B,*+

C,

is commutative. If a E A , , then r i a i = a for some i E Z, and so $,$*a = $,4*7ciai= $ , p i 4 i a i = 0 , $ ~ 4 ~ a , = OLet . b E K e r $ , . For some b i € B i , p i b i = 6, whence c i $ i b i = $ , p i b i = $,b = 0. Therefore, for some j 2 i, oj$ibi = 0, and so $ j p j b j = ajt,bibi = 0. The top row in the last diagram is exact, therefore there is an a j E A j with 4 j a j = p j b i . Setting a = 7cjaj, we arrive at $*a = $ , n j a j = p j 4 j a j= p j p j b i = p i b i = b, i.e., b E Im 4, and the bottom row is exact.0 If we apply the second statement of (1 1.2) and (1 1.3) to the case when the $ i are epimorphisms, then we are led to:

4; are monomorphisms and the Corollary 11.4. and, in addition,

If A , B, C are direct 0 --t A ; -+Bi 4i

systems as in the preceding theorem

%-=

C; 4 0

is exact f o r ecery i E Z, then the induced sequence f o r the direct limits

O-+ A,-%B,

*+c,+o

is likewise exact.m This statement can be expressed less accurately, but more lively, by saying that a direct limit of exact sequences is exact. [In other words, the functor L , : 2, + d is exact.]

EXERCISES 1.

(a) Let A , E'Z ( n = 1 , 2, .*.), and let n:" tion by n. Prove that

: A , - + F I , ~ +be ~ multiplica-

LIJ A , E Q . (b) A group is locally cyclic if and only if it is a direct limit of cyclic groups.

12.

INVERSE LIMITS

59

2. If A , is the direct limit of the system A = { A i( i I )~; n{},then for any integer m > 0, m A , is the direct limit of the system {mA,( i E I ) ; n{ I M A , } . 3. (a) If A , is the limit of the direct system A = { A i(i E I ) ; xi}, and if a E A , , then there exist an i E Z and an a, E A i such that n i u i = a and o(u,) = o(a).

(b)* If c( : G -+ A , where G is finitely generated, then there exists an i E I and an a i: G -+ A , such that CY = r i a i .[Hint: (15.5).] 4. The direct limit of torsion [torsion-free] groups is again torsion [torsionfree]. 5 . Let A = { A , (i E I ) ; ni} be a direct system with limit A , and with canonical maps n i : A i+ A , . Define A , = A , and .,'" = n iwith i < co for all i E I , and show that A' = { A , ( i E I u {a})ni} ; is a direct system with limit A , . 6. If A = { A i(i E I ) ; nj} and B = { B i(i E I ) ; p i } are direct systems of groups, then { A0 ~ B~( i E I ) ; 7rj 0p i }

7.

8. 9.

10.

is again a direct system whose limit is the direct sum of the direct limits of A and B. Give an example showing that the direct limit of splitting exact sequences need not be splitting. [Cf. (29.5). Why does this not contradict Ex. 6?] Show that the direct limit of longer exact sequences is likewise exact. What is wrong with the following argument? Using Example 3, on page 56, the sequence0 -+ Z(p") -+ Z(p") -+ Z(p") -+ Ocan be represented -+ Z(p")-+ 0, as the direct limit of exact sequences 0 -+ Z(p")-+ Z(p2n) so it must be exact because of (1 1.4). Give an example to show that a subfunctor F of the identity need not commute with direct limits. 12. INVERSE LIMITS

Next we turn to the discussion of inverse limits; these are, in a certain sense, dual to direct limits. Assume A (i E I ) is a system of groups, indexed again by a directed set I, and for each pair i, j E I with i 5 j there is given a homomorphism

ni:A j -+ A , such that

( i 5.i)

is the identity map of A i , for each i E I, (i) (ii) for all i s j 5 k in Z, we have n{ n; = n!.

60

11. DIRECT SUMS

Then the system A

=

{Ai( i E I ) ; i$)

is called an inverse system. The inverse [or projective] limit of this system,

A* = U , A , , is defined to consist of all vectors a = (. .. , a, , . . .) in the direct product A = Hi., A ifor which

n!a. = a.,

(i5.j)

holds. A* is a subgroup of A , for if a, a’ E A*, then their coordinates satisfy n ~ a j = a , , n ~ a J =t ah u~ s; , n ~ ( a j - a J ) = a , - a ~ a n d a - a ‘ E A * . For inverse limits the following simple facts are worthwhile mentioning. (a) There exist homomorphisms n, : A*

+A i

(iE I )

such that the diagrams A* (1)

are all commutative. In fact, n, : a w a i [i.e., the restriction of the ith coordinate projection of A , to A*] satisfies this condition. These n iwill be called canonical. (b) If every ni is a monomorphism, then all the n, are monomorphisms. Assume are monic, and let nia = 0 for some a E A*, i E I. Given j E I, let k E I satisfy i, j 5 k . Now nk nka = x ia = 0, whence, by our hypothesis, n k a = 0. Therefore, n j a = $n,a = 0 for every j , and so a = 0. (c) If K is a cojnal subsystem of I, then

n

@K

A,

b, A,.

To a’ E b,A , 5 n k s K A k there , exists a unique a E b, A , such that, for every k E K , the kth coordinates of a’ and a are equal. In fact, if we define a = ( . . *,a i , -..) with a , = n!a; (i 5 k ) , then a ’ w a is an isomorphism between the two inverse limits. (d) A* is the intersection of kernels of certain endomorphisms of I Ai. Let (i,j ) be a pair of elements of I satisfying j 2 i. It gives rise to an endomorphism of the direct product Ai :

Hi

fl

f+i,j)(--*, a,,

..., a j , . . - ) I + ( . . .

, 0 , a i - n:aj, 0, ...),

12.

INVERSE LIMITS

61

where at most the ith coordinate is not zero. If we compare this with the definition of inverse limits, it is evident that

with ( i , j ) running over all pairs of the stated kind. (e) If all the groups in the iniierse system A = { A i (i E 1 ) ; xi} are (Hausd o r f ) topological groups and all the n! are continuous homomorphisms, then the inverse limit A* is a closed subgroup of n A i [the latter group is equipped with the product topology],and the canonical maps n, : A* + A iare continuous. If (..., a , , ..-)E A i is not in the inverse limit A*, then there exist i < j in Z such that a, # n i a j . By the Hausdorff property, there are disjoint open subsets U , Vin A , with a, E U and n {a jE V . Now the set of all (-.., b , , ...) E A , with bi E Uand n!bj E V is an open set in A i that contains (. . . ,a i , * . .) and is disjoint from A*. The last assertion follows from the continuity of the coordinate projections. We have the dual of (11.1):

n

fl

Theorem 12.1. The inverse limit A* of the inverse systen A = { A , (i E 1 ) ;xi} satisfies: if G is a group and if there are homomorphisr 7s a,; G -+ A , with commutative diagrams G

then there exists a unique homomorphism a : G + A* for wd ich all the diagrams G L-+ A*

are commutative [where n i is the canonical homomorphism]. This property characterizes A* and x i up to isomorphism.

n

For g E G set ag = (. . . , Gig, . . .) E A , . Because of the commutativity of (2), og E A*. Thus a : G -+ A* is a homomorphism satisfying a,g = niag, whence the commutativity of (3) results. If a’ : C + A* also makes (3) commutative, then n,(a - a’) = 0 for every i. Hence, for g E G every coordinate of (a - 0’)svanishes and thus - a‘ = 0. In order to establish the second assertion, assume A , and maps T~ : A , + A i have the property formulated in the first statement of the theorem. Then there exist unique homomorphisms a : A , + A* and a, : A* + A , satisfying x ia = T~ and T~ a. = n i. We infer n, = x iaao for every i, and hence ano is the

11.

62

DIRECT SUMS

identity map of A*. This shows &,A* a direct summand of A , . From the uniqueness of G + A , for every G, we conclude uoA* = A , , and u, is an isomorphism. 0 Example I . Let

be the direct product of groups B, . Define J to consist of all finite subsets ctof the index set I where ct 5 /3 means '' ct is a subset of p." For ct E J let A. = 0,Biand for OL 5 p let rf denote the projection map A, + A , [i.e., we drop the coordinates with indices in p not in a]. Then = limJ

A*

A, g A.

f-

To prove this, let re denote the canonical maps in (1) and us the projections A + A , . By (12.1) there is a unique u : A + A* such that r au = ua. If ua = 0 for some a E A, then u. a = v auu = 0 for all ct E J , and so u = 0, u is a monomorphism. If a* = (. . . ,a,, . . .) E A * , then write a. = b,, . . . b,, (bi E B i ) , where ct = {i,, . . . , i k } .Because of the choice of ri,il E implies that the ilth coordinate of up must be bi,, hence a* defines a unique (. . ' , b i , . . .) E A . A glance at the definition of u in the proof of (12.1) shows that u(. . , bi , . . .) = a*, so u is a n epimorphism, and hence a n isomorphism.

+ +

be a cyclic group of order p" (n = 1,2, . . .), and let Example 2. Let C, = (c.) C.+,-C. act as T ~ + * C , +=c,. ~ The meaning of r . " ( m L n ) is then obvious. Now {Cn ( n = 1 , 2, .. .); v."}is an inverse system such that

T;+I:

C*

= lim,

c

C,

J, .

If r ndenotes the canonical map C * + C, , and if we define u. : J , + C. by on1 = c, (1 E J,), then there is a unique u : J , 4 C* such that rnu = u, . Since no element # O of J p belongs to every Ker o n ,Ker u = 0. If c = (c;, . . . , c:, . ..) E C * with c: = k , c . ( k . E Z), then by the choice of r;+' we have k , + , = k , modp", and there is a p-adic integer T such that T = k, mod p" for every n. Thus U,T = c: , and u must be epic.

niEI

A , . Define Example 3. Let A, ( i E I ) be subgroups of a group A and A , = i < co for all i E I, and i 5 j for i , j E I u {a)}to mean that A j 5 Ai and let r{ be the injection map A j + A , . Then

A

=

{ A i ( i e I u {co});

is an inverse system with inverse limit A , . In fact, A* = lim A consists of all vectors in c A i whose coordinates are the same element of A , . This example shows that intersections may be regarded as inverse limits.

The notion of homomorphism for inverse systems can be defined analogously to direct systems. Let A=(Ai

(iEZ);

n;}

and

B = { B i ( ~ E Z ) ;p i }

be inverse systems, indexed by the same directed set I. A homomorphism J#C : A + B is a set of homomorphisms {$i : A i + Bi ( i E I ) } subject to the condition that the diagrams

12.

63

INVERSE LIMITS

Theorem 12.2. If 4 : A -+ B is a honiomorphism between the inuerse systems A , B, then there exists a unique liomomorphism @*: A* =& I A i B* = ]im, Bi such that, .for every i E I , the diagram -+

is commutative [n,,p i denote the canonical maps]. If every 4 , is monic, then q5* is a monomorphism. The homomorphisms 4, (i E I ) induce a homomorphism The commutativity of (4) implies that if a = (. .. , a , , . . .) E A*, then its image under 4 is contained in B * ; hence define 4* : A* -+ B* as the restriction of 4 t o A * . With this #*, c # ~ ~ = n ~4a, n i = p i 4 * a , and ( 5 ) is commutative. If c$o : A* -+B* makes (5) commutative for every i, then p i ( 4 * - 4,,) = 0 for every i, thus all the coordinates of (4* - &)a vanish, for every a E A*, and so 4* - $ 0 = 0. Jf every 4, is a monomorphism, and if $*a = 0 for some a E A*, then $ Jn, ~ a = p i 4*a = 0 implies xia = 0 for every i E I , whence a = 0 . 0 For a fixed directed set I , consider the category Y , of all inverse systems indexed by I with homomorphisms as defined before (12.2). The function L* that assigns to an inverse system A its inverse limit A* and to a homomorphism C$ : A + B the homomorphism C$* : A* + BY is-as is readily seen from ( I 2.2)-a functor o n 3 , to d.Hence we may refer to C$* as a natural homomorphism between the inverse limits.

For the inverse limit of exact sequences we have a weaker result than for direct limits. Theorem 12.3. Let

A = { A i( i E I ) ; xi}, B = {Bi ( i E I ) ; p i } , C = (C, (i E I); CJ;} be inverse systems with the same index set I, and 4 : A -+ B, $ : B C homomorphisms such that the sequence o-+A-%B-+c -+

*

64

11.

DIRECT SUMS

is exact [i.e.,

0 -+ /Ii&

Bi&

ci

-

is exact for every i]. Then for the inverse limits we have the exact sequence O + A * 4' B* d'* C* where 4*, $* denote the maps described in (12.2). The exactness at A* is just the second statement of (12.2). From the definition of +*, $* it is evident that $*$* = 0. If n i , p i , cri are the canonical maps, then the diagram 0- A* &B* C*

x+

is commutative. In order to show that the top row is exact at B*, let b E Ker $*. In view of t,hi p i b = cri $*b = 0 and the exactness of the bottom row, there is an ai E A i , for every i E I , for which 4iai = p i b . For j > i, +injaj = p i 4 j a j = p { p j b = p i b = + i a i , whence .{aj = a i , 4i being monic. We infer that a = (* * * ,ai , . * . , a j , .) E A*. For this a we have p i 4*a = +inia = +iai = p i b for every i, and so +*a = b . 0 Let us note that by (12.3), the functor L*: X I+ d is left exact.

Exercise 6 will show that (12.3) cannot be improved by putting +O at the end of the displayed formulas.

EXERCISES 1. Let A , E' Z(n) = (a,), and for n I m, let nr : A , + A , be the homomorphism that maps a,,, upon a,, . Then { A , (n E I ) ; nr} is an inverse system [I is partially ordered by the divisibility relation] whose inverse limit is

n,

JP.

: A,,,, + A , be the multiplication by p. Then 2. Let A, Z(pm)and let n:" the inverse limit is the group of all p-adic numbers. 3. If A = { A i (i E I ) ; ni} and B = {Bi ( i E I ) ; p i } are inverse systems, then so is { A0 ~ B~( i E I ) ; ni GI p i } ,

and its inverse limit is the direct sum of the inverse limits of A and B. 4. The inverse limit of torsion-free groups is torsion-free, but the inverse 5.

limit of torsion groups can be torsion-free #O. If in an inverse system { A i (i E I ) ; ni}, all the maps ni are isomorphisms, then all the ni are likewise isomorphisms.

13.

65

COMPLETENESS AND COMPLETIONS

6. Let A , = Z and 7~: = 1, for all positive integers n, m ( 2 n). Let B, = Z(p") and let p," carry 1 of Z(p") into 1 of Zcp"). Show that (a) {A,,; x,"} and { B , ; p,"} are inverse systems such that 4 = {4,,} with $" : A , + B, the canonical map is an epimorphism; but (b) the induced homomorphism 4* between the two inverse limits is not epic. 7. Show that the inverse limit of splitting exact sequences need not be splitting exact. 8. Let A = { A , (i E I ) ; xi} be a n inverse system with limit A* and canonical maps xi: A* -+ A , . Define A , = A* and x y = xiwith i < 00 for all i E I, and show that

A'

=

{Ai(iE I u {co};ni}

is an inverse system with the same limit A*.

13. COMPLETENESS AND COMPLETIONS

Abelian groups which are complete in some topology will be of importance for us. Therefore, in this section, we wish to examine completeness and completion processes. All the topologies to be considered here are linear. Assume a linear topology is defined on the group A in terms of the dual ideal D of L(A). We label the subgroups U in D by an index set I, and put i 5 j for i, j E Z to mean U , 2 U j . Thus I-as a partially ordered set-will be dual-isomorphic to D, and hence I is directed. As usual, by a net in A we mean a set { a , } , € of elements a, E A , indexed always by Z; in other words, a net is a function on I to A . A net {a,}, is said to converge to the limit a E A if to every i E I there is a j E I such that ak - a E U , whenever k 2 J. If A is Hausdorff in the given topology, then limits are unique; if, however, A fails to be Hausdorff, then limits are determined only up to mod U , . The classical proof applies to show that a subgroup B of A is closed exactly if it contains the limits of convergent nets with members in B. Evidently, if { a i } , € is convergent, then so is every cofinal subnet, and it has the same limit. To facilitate discussion, we shall concentrate on nets { b , } , , , such that, for every i E I , b, - a E U , for all k 2 i [i.e., j = i may be chosen in the definition of convergence]. We shall call such a net neatly convergent. A net {a,}, is called a Cauchy net if to any given i E I there is a j E I such that

0,

ak - ak,E U ,

whenever

k , k' 2 j .

66

11.

DIRECT SUMS

Since in the present case U , is a subgroup, ak - a j , a k ,- a j E U,already implies ak - ak.E U i , so that, for the Cauchy character of {a,}, it suffices to know that a, - a j E U iholds for all k Z j . Clearly, every cofinal subnet of a Cauchy net is again Cauchy, and if a cofinal subnet of a Cauchy net converges, then the whole net converges to the same limit. We shall consider Cauchy nets {bi}i which are neat, in the sense that for every i E I, EI

6, - bi E U i

whenever k 2 i .

I f a neat Cauchy net {bi}is t converges to a limit a, then it neatly converges to it; for if b, - b E U i for some j which may be assumed to be L i , then b, - bi E U i implies b, - b E U i , and hence b, - bi E U i (k 2 i) implies bk - b E Uifor all k 2 i. A group A is called complete in a given topology if it is Hausdorff, and every Cauchy net in A has a limit in A . An equivalent definition may be given by restricting ourselves to neat Cauchy nets. [Notice that, unlike many authors, we have defined completeness for Hausdorff spaces only.] We have the rather simple:

Proposition 13.1. Let the group A be complete in a topology. A subgroup of A is closed if and only if it is complete in the induced topology. For the proof we refer, e.g.,to J. L. Kelley,“General Topology”(1955).0 In the next result the countability hypothesis is essential.

Proposition 13.2. I f B is a closed subgroup of a complete group A satisfying the first axiom of countability, then the quotient group AIB is complete in the induced topology. By hypothesis, there is a base { U , } , of neighborhoods of 0 in A such that U , 2 U , 2 * . and Urn= 0. The groups Urn= ( U , + B)/B ( m = 1,2, * . -) form a base of neighborhoods of b in AIB. Let a, B, * * . , a, B, . be a Cauchy sequence in A / B which may, without loss of generality, be assumed to be neat, i.e., a,,, - a, + B E Urn+ B for every m. Define c, = a,, andassume we have chosen cl, . .. , c, E A such that c, E ai + B and ci - c i - l E Ui- for i = 2, * , m. Then a,+, - c, = u, b, for some u, E Urn,b, E B, and we set c,+ = a,+, - b, E a,,, + B to have c,,, - c, E U , . Therefore, there.is a sequence {c,}, such that c, E a, B and c,+, - c, E U,,, for every m. In other words, the given Cauchy sequence {a, + B } in AIB can be lifted to a Cauchy sequence {c,} in A . If a E A is the limit of {c,}, then a = a + B is the limit of {a,,,+ B } . Since B is closed in A , AIB is Hausdorff.0

0,

,

+

+

,

+ +

Let A , ( t E J ) be a family of groups, each equipped with a linear topology, say, defined in terms of the dual ideal D, of L(A,). Let G = A , be the direct product of these groups and n,the tth coordinate projection, n,: G -+ A,.

n

13.

67

COMPLETENESS AND COMPLETIONS

Recall that the product topology on G is defined such that a subbase of neighborhoods of 0 is the set of all n; U , with U , E D, and t E J . This is again a linear topology, and the n, are continuous, open homomorphisms.

fl

There is another topology on the direct product A , which will be most important for us. Without loss of generality, one may assume that the very same index set Z serves to index a base of neighborhoods about 0 in each A , : { U , i } i e in l A , . [Notice that j 2 i in Z implies U t i2 U I jfor every t, but U I i= U , j may occur for i # j.] The box topology on G = A , is defined to have the subgroups

n

JJ u,i= u: f

(i

E

I)

as a base of neighborhoods about 0. This topology is linear; it is Hausdorff exactly if all the A , are Hausdorff and satisfies the first countability hypothesis if all the A , do. The inclusion U: 2 n;' U,i (for every t) shows that the box topology is finer [i.e., has more open sets] than the product topology. Hence the n, are again continuous and open. Notice that the box topology may depend on the way the U l i are indexed. An important special case is when all the groups A , have their Z-adic [p-adic] topologies. Then the box topology in A , is just the Z-adic [p-adic] topology of its own. The box topology has the advantage that in the direct product we may use the same set I to index nets. An elementary calculation shows that, in the box topology of G = A , , the projection { n , g i } i e Iof a (neat) Cauchy net { g i } i E in I G is a (neat) Cauchy net in A , , and a net { g i } i e Iis a neat Cauchy net if, for every t E J , { n , g i } i e ris a neat Cauchy net in A , .

n

Proposition 13.3. A direct product is complete in the box topology if and only i f every component is complete. Let G = A , be complete in the box topology, and let be a Cauchy net in A ,. Then { p , ai} I is a Cauchy net in G [p, denotes the tth coordinate injection], say, tending to the limit g E G. Evidently, n,g is the limit of {a,}. Conversely, assume all the A , complete, and let { g i } i E Ibe a neat Cauchy net in G . If a, E A , denotes the limit of the Cauchy net {n,gi}i and if g E G is defined so that n,g = a , , then it follows at once that g is the limit of {Si>iE 1 . 0

n

If all the groups A , are furnished with the 2-adic [p-adic] topology, then we have : Corollary 13.4. A direct product of groups is complete in theZ-adic [p-adic] topology if and only i f each component is complete in its Z-adic [p-adic] topology.

11.

68

DIRECT SUMS

In the rest of this section we turn our attention to the problem of embedding a group A , equipped with a linear topology, in a complete group 2.We discuss two methods, one based on Cauchy nets and another on inverse limits. Let A be a topdlogical [not necessarily Hausdorff] group and {Ui}iel.a base of neighborhoods of 0. A net { u ~ I} in~ A can be identified with an element of the direct product G = A' under the natural correspondence (a,},. ' H(- *

*

,a,, . .). *

In order to avoid unnecessary complications, we shall restrict ourselves to neat Cauchy nets; this can be done without loss of generality. It is readily checked that the neat Cauchy nets in A form a subgroup C of G, and the nets neatly converging to 0 form a subgroup E of C. It follows at once that E is closed in C if Cis furnished with the product or with the box topology; hence 2 = C / E is a Hausdorff topological group in either induced topology. Lemma 13.5. The product and the box topologies of A' induce the same topoIogy on A. Since the two topologies are comparable, it suffices to show that to every neighborhood U: in the box topology there is a neighborhood n,Y1Uk in the product topology for which

+

(C n nY1Uk) E 5 (C n U:)

+E

holds. By making use of the modular law, this amounts to If { a l } l I is a neat Cauchy net contained in 7 ~ ;' U i ,then not only a, E U , holds, but also a j E U , for a l l j 2 i, whence { u ~ } E~ U? E . 0

+

An obvious consequence of this lemma is that a base of neighborhoods of 0 in A is obtained by forming the subgroups 0, of A, consisting of all neat that a, E U , for all 1. Cauchy nets { u ~ } such ~ The mapping p : U H ( . * * , a, - . * ,a, E * a * )

+

of A into 2 is clearly a continuous, open homomorphism. Moreover, if A is Hausdorff, p is a topological isomorphism between A and p A if p A is taken in the induced topology of A [this topology of A is described in (13.5)]. Theorem 13.6. The group 2 is complete and p A is a dense subgroup of 2. Given (..., a,, E E 2 and a neighborhood U: of 0 in G, it is clear that pa, lies in the U,*-neighborhood of (-..,ai , . * -) + E. Hence p A is dense in A, and therefore, to prove completeness, we need only verify the convergence of Cauchy nets in p A to elements of A. A neat Cauchy net in p A is of - a * )

+

13.

69

COMPLETENESS AND COMPLETIONS

the form {pai}i,I where { a i } i e is l a neat Cauchy net in A . It is straightforward to check that (. . * , a , , - ..) + E is the limit in A^ of the net { p a , } , .O Another method of constructing completions is via inverse limits. Starting again with { U i } i s l ,let us define Ci = A / U i and, for j 2 i in 1, the homomorphism ni : C j +Ci as ni(a + U j ) = a + U i.Then we get an inverse system C = { C , ( i E I ) ; n:} whose inverse limit we denote bye. Observe that the groups Cicarry the discrete topology, and is considered as equipped with the topology induced by the product topology on Ci . Thus a base { V,} I of neighborhoods of 0 in is given by the subgroups n n,: ‘0 [n,denoting the ith coordinate projection]. Clearly,

c

c

a : a + + ( - - . ,a

ne

+ ui,

e.0)

E

c

is a homomorphism which is Gontinuous and open. Proposition 13.7. There is a natural map 1 : A^ -+ which is a topological isomorphism such that Ap = a. If { a i } i is a neat Cauchy net, representing an element of A^, then we define

e

1:( . * . ,a , , . . * ) + E H ( . - * a, , + U , , --.)

c

+

+

+

where the image is in since ni(aj U j ) = uj U , = a, U , for j 2 i. If * ,a , + U i , . * -)E e , then { a , } i I is a neat Cauchy net, and another selection of representatives yields the same coset mod E . That 1 maps neighborhoods upon neighborhoods is obvious, and so is Ap = a . 0 (a

-

c

Since we wish to speak of A^ z as the completion of A , a unicity statement is necessary. This will follow from the following universal property of A^. Theorem 13.8. Given a continuous homomorphism 4 of A into a complete group G, there exists a unique continuous homomorphism : A^ + G such that 6 P = 4. Let {a,}, I be a Cauchy net in A with limit E A^. Then continuity implies that { & z , } , ~must ~ be Cauchy in G, and if g E G is its limit, then the only possible way of defining a continuous is to put : a H g . The rest of the assertion is self-evident.0

6

6

From (13.8) it follows that A^ is unique up to topological isomorphism. Also, p : A A^ is a natural map, for if c 1 : A + B is a continuous homomorphism, then the diagram A”-B --f

commutes where the meaning of p’ is obvious and B is the homomorphism whose existence is stated in (13.8).

11. DIRECT SUMS

70 EXERCISES

1. Prove that pl [with p as defined in the text] is topologically isomorphic to AIU, where U is the intersection of all open subgroups of A . 2. Every compact group is complete. 3. The inverse limit of complete groups is complete. 4. If A has a base of neighborhoods about 0 consisting of subgroups of finite index, then the completion of A is compact. 5. Find the completion of Z in the p-adic [and in the Z-adic] topology. 6.* ( S . Lefschetz) A group A is called linearly compact if it has a Hausdorff linear topology such that, if C j are closed subgroups of A and any finite number of the cosets a j C j (aj E A ) have a nonempty intersection, then the intersection of all a j + C j is not vacuous either. Prove: (a) In this definition, " closed " can be replaced by " open." (b) If A is linearly compact and if a : A + B is a continuous epimorphism, then B is linearly compact. (c) If A is linearly compact, then every closed subgroup of A has this property. (d) Direct products and inverse limits of linearly compact groups are linearly compact [in the product topology]. (e) Linearly compact groups are linearly compact in any coarser topology. 7.* (a) Linearly compact groups are complete. (b) A group is linearly compact in the discrete topology if and only if it satisfies the minimum condition on subgroups. (c) Give examples for a complete group which is not linearly compact and for a linearly compact group which is not compact.

+

NOTES Most of the results of Chapter I1 could have been stated more generally for certain categories;for a systematic treatment of these questions we refer to B. Mitchell[Theory of Categories (1965)l. Theorems (8.1), (8.2), ( l l . l ) , and (12.1) point out the universal [and co-universal] properties of the concepts under consideration. The universality of certain objects is essential in general category theory. The primary decomposition theorem (8.4) is of central importance (in the finite case, such a decomposition characterizes the nilpotent groups). E. M a t h 1 Trans. Amer. Math. SOC.125,147-179 (1966)] considered commutative domains R such that every torsion Rmodule M is a direct sum of submodules Mp where for every a E M p , o(a) is contained in exactly one maximal ideal P of R. He proved that R has this property if, and only if, every nonzero ideal of R is contained in but a finite number of maximal ideals and every prime ideal # O in exactly one maximal ideal. In contrast to (8.4), (8.5) easily generalizes to arbitrary modules: if a module is the union of simple submodules, then i t is a direct sum of simple modules [it is then called a semisimple module]. Semisimple modules may be characterized by the condition that every submodule is a direct summand.

NOTES

71

Problem 3. (a) (J. Irwin) For which subgroups B of a group A are all B-high subgroups isomorphic ? (b) The same question with the B-high subgroups restricted to those containing a maximal fully invariant subgroup of A disjoint from B. Problem 4. Give an upper bound for the number of nonisomorphic B-high subgroups of A , in particular, if B = A ' . P r o b h 5 . Investigate the groups in which every infinite subgroup B can be embedded in a direct summand of cardinality IB(. See Irwin and Richman [l].

Problem 6. Which classes of abelian groups are closed (a) under taking subgroups and direct limits? (b) under epimorphic images and direct limits? E.g., locally cyclic groups. Naturally, analogous questions can be asked for other operations on classes; cf. 18, Ex. 7.

Problem 7. (B. Charles) Describe the functorial subgroups that commute with direct products [or inverse limits]. Problem 8. Which groups can be complete in some metric (linear) topology ? In this connection see 39.

I11 DIRECT SUMS OF CYCLIC GROUPS

In this chapter we begin the study of important classes of abelian groups. First we shall be concerned with direct sums of cyclic groups. Their importance stems from the fact that they can easily be characterized by satisfactory invariants, and the analysis of other classes of abelian groups is based, to a certain extent, on our knowledge of direct sums of cyclic groups. The first section is devoted to free groups and to a brief discussion of defining groups in terms of generators and defining relations. This is followed by a description of finitely generated groups; the so-called Fundamental Theorem is referred to in various branches of mathematics. For infinitely generated groups, one can establish criteria under which a group decomposes into a direct sum of cyclic groups; these are, however, only easy to handle in the torsion case. One of the most useful results asserts that the class of direct sums of cyclic groups is closed under passage to subgroups. Every abelian group A contains subgroups that are direct sums of cyclic groups. Those which are, in a certain sense, maximal among them, define cardinal numbers depending only on A. This leads to the definition of ranks which are very useful invariants for A.

14. FREE ABELIAN GROUPS-DEFINING

RELATIONS

By a free abelian group is meant a direct sum of infinite cyclic groups. If these cyclic groups are generated by elements x i (i E Z), then the free group F will be

Thus, F consists of all finite linear combinations g = nlxi,

+ + nkxik

14.

FREE ABELIAN GROUPS-DEFINING

RELATIONS

73

with different x i l ,..., x i k where nj are integers f O and k is a nonnegative integer. In view of the definition of direct sums, two linear combinations ( I ) are equal elements in F if and only if they differ at most in the order of summands, and the addition of two linear combinations is to be performed formally, by adding the coefficients of the same x i . Clearly, we could have equally well defined F by starting with a nonempty set of symbols x i , called a free set of generators, and then declaring F as the collection of all formal expressions of the form (1) under the mentioned equality and addition. In view of this, F is also called the free group on the set X . It is evident that the free group F is, up to isomorphism, uniquely determined by the cardinal number in of the index set I . Thus we are justified to write F,,, for a free group with m free generators. Now we have the following simple result. Proposition 14.1. The free groups F,,, and F,, are isomorphic if and only if for the cardinals rn, n. We need only verify the “only if” part of the assertion. Let p be a prime and F a free group with m free generators x i . Since every element g E F has a unique form (l), it is clear that g E ~ isF equivalent to the simultaneous fulfillment of the divisibility relations p In,, . . . , pI nk . Hence F/pF-as a vector space over the prime field of characteristic p-has a basis { x i + pF}; thus, its dimension is m. The dimension being an invariant of vector spaces, the assertion follows. 0

m

=n

Consequently, there is a one-to-one correspondence between nonisomorphic free groups and the cardinal numbers. If in is the cardinal corresponding to the free group F, we call F ofrank m. The rank of a free group F is uniquely determined. One of the basic properties of free groups is contained in the following result. Theorem 14.2. A set X = { x i } i s Iof generators of agroup F is a free set of generators [and hence F is a free group] if and only if every mapping 4 of X into a group A can be extended to a (unique) homomorphism $: F --* A . Let X be a free set of generators of F. If 4 : x i - a i is a mapping of X into a group A , then define $: F + A as $(n,xi,

+ ... + n k x i k )= n l a i , + . * .+ n k a i k .

The uniqueness of (1) guarantees that II/ is well defined, and it is readily checked that it preserves addition. Conversely, assume that the subset X in F has the

74

111. DIRECT SUMP, OF CYCLIC GROUPS

stated property. Then let G be a free group with a free set {yi}i I of generators, where I is the same as for X . By hypothesis, 4 : xi b y i (i E I ) can be lifted t o a homomorphism @ : F + G, which cannot be anything else than the map $ : nlxil + . . . + nk xirw n l y i l + . . . + nk y i , . It is evident that II/ must be an isomorphism. 0 In particular, mapping X onto a generating system of A , we arrive at the following result.

Corollary 14.3. Every group with at most image of F,

.o

111

generators is an epimorphic

For an infinite cardinal lit, F, has 2" subsets, and hence at most 2" subgroups and quotient groups. We infer that there exist at most 2" pairwise nonisomorphic groups of cardinality m. The following is a n elementary, frequently used property of free groups.

s

Theorem 14.4. I f B is a subgroup of A such that A / B is free, then B is a direct summand of A. By (9.4) it suffices to verify this for the case in which A / B is a n infinite cyclic group, say A / B = ( a * ) . Select a representative a E a* in A. Then the cosets na* (n = 0, I , + 2 , . . .) mod B are represented by the elements na of ( a ) . Hence A = BO ( a ) . O The next theorem provides complete information about the structure of subgroups of free groups. Theorem 14.5. A subgroup of a free group is free. Let F = O i G (Ia i ) be a free group, and suppose that the index set I is well ordered in some way; moreover, I is the set of ordinals a.For r~ z we define F, = O p < ,(a,). If G is a subgroup of F, then set G, = G n F,. Clearly, G, = Gofln F,, so G,+I/G, z (G,,, + F,)/F,. The latter quotient group is a subgroup of F,+,/F, z (a,); thus either Gofl= G, or G,+,/G, is an infinite cyclic group. By (14.4) we have G,+l = G, (b,) for some 6, E G , + , [which is 0 if G,+, = G,]. It follows that the elements 6, generate the direct sum (b,). This direct sum must be G, because G is the union of the G, .O

s

There is a concept closely related to freedom which is-in the case of abelian groups-equivalent to it [but this is not so for modules in general]. Call a group G projective if every diagram

G

14.

FREE ABELIAN GROUPS-DEFINING

75

RELATIONS

with exact row can be completed by a suitable homomorphism $ : G + B to a commutative diagram. Since in this case C is essentially a quotient group of B , the projectivity of G can be formulated in other words by asserting that every homomorphism of G into a quotient group of any group B can be lifted to a homomorphism of G into B. Now we have:

Theorem 14.6 (Mac Lane [l]). A group is projective if and only if it is free. Let p : B + C be a n epimorphism and F a free group with 4 : F -+ C . For each x i in a free set {xi}iEI of generators of F, we pick out some bi E B such that p b i = 4 x i , which is possible, p being epic. The correspondence x i H bi (i E I ) can, owing to (14.2), be extended to a homomorphism $ : F-+ B. This $ satisfies /?$ = 4; thus F is projective. Let G be projective and fl : F -+ G an epimorphism of a free group F upon G. Then there exists a homomorphism $ : G + F such that p$ = 1., Hence $ is a monomorphism onto a direct summand of F, i.e., G is isomorphic to a direct summand of F. By (14.5), G is f r e e . 0 By (14.3), for every group A we can find an exact sequence O+H+F+A+O with F, and hence by ( 1 4 3 , H free. This is called a free or projective resolution of A . The results of this section have numerous consequences; in fact, we shall see that they are often referred to. They have been generalized in various directions. The case of modules over the p-adic integers will be of importance to us; let us therefore formulate the next theorem.

Theorem 14.7. The results in ( I 4.1)-( 14.6) prevail i f abelian groups are replaced by p-adic modules. In fact, the proofs hold verbatim with the obvious changes.0 The results of this section enable us to define a group in terms of generators and defining relations. Though this method is well known from general group theory, we dwell upon this subject for a little while. Let A be a group with a set { a i } I of generators, and let q : F + A be a n epimorphism of a free group F = B i t [ ( x i ) upon A , such that qxi = a i . Obviously, Ker q consists of all elements m l x i , + . . . m k x i kE F ( m i E Z) such that m l a i , + . . . + mkai, = 0 in A . [Equalities of this type are called defining relations relative to the generating system { a i } i I of A . ] Bearing this in mind, one can define the group A as

+

+ +

A = ( a i ( i E I) ; rnj uiI . . . rn j k a,, = 0 ( jE J ) ) , (2) meaning thereby the group A = F / H where F is a free group, generated by the free generators x i ( i E I ) , and H is the subgroup of F, generated by the

76

111.

DIRECT SUMS OF CYCLIC GROUPS

+

elements mjl x i , . . . + m j k x i rof F, for all j E J , corresponding to the left members of the defining relations. This definition is standard. (2) is called a presentation of A . [One should, however, keep in mind that a group can be presented in various ways as a quotient group of a free group.] Considering that H is, by (14.5), again free, H = @ k E K ( Y k ) , where every y k may be written in the form yk

=

is1

nkiXi

(nki

E

z, k E K ) ;

for a fixed k , almost all nki vanish. Thus the presentation A row-finite matrix (3)

IlnkillkEK,isI

= F/H

defines a

(nki

with independent rows. Conversely, every row-finite matrix with independent rows gives rise, in the obvious way, to a group A . Since a group can be represented in several ways as a quotient group F/H of a free group F, and since one can write F and H in various ways as direct sums of cyclic groups, different matrices of the form (3) may yield isomorphic groups. One of the major problems in this connection is to find out when two matrices define isomorphic groups. It is not hard to state conditions for.this, but they do not help to decide isomorphy and have no practical application so far.

EXERCISES 1. A generating system of F, , m a positive integer, is a free set of generators, if and only if it contains exactly m elements. 2. Prove the following converse of (14.4): if F is a group such that B 5 A and A / B E F imply that B is a direct summand of A , then F is free. 3. (Kerttsz [2]) F is free if F is isomorphic to a subgroup of any group G, whenever there is an epimorphism G + F. 4. (Kaplansky [2]) (a) Let F be a free group, G a subgroup, and H a direct summand of F. Then G n H is a direct summand of G. [Hint: G/(G n H ) is free.] (b) The intersection of a finite number of direct summands of a free group is again a direct summand. 5 . If F is free and : F -+ A is an epimorphism with A finitely generated, then there is a direct decomposition F = Fl @ F2 such that Fl is finitely generated and F, 5 Ker q. 6. If A is presented by a set of generators and defining relations and B by a subset of these generators and definingrelations, then there is a natural homomorphism B + A , by letting the generators of B correspond to themselves qua elements of A . 7. Let A be presented by a set of generators and defining relations and

15.

8.

9.

10.

77

FINITELY GENERATED GROUPS

assume that the generators can be divided into two classes { b i } i E I , { c j } j E Jsuch , that every given defining relation contains only bi or only c j terms explicitly. Then A = B 0C, where B is generated by the bi and C by the c j . To every system of generators of a group, there is a set of defining relations relative to these generators, such that no relation can be omitted. [Hint: use (14.5).] Let F be a free group, G a subgroup of F, and A a group with the property that every 4 : G --f A can be extended t o a $ : F - t A . Prove that the same holds if A is replaced by a n epimorphic image of A . Let B be a subgroup of A , and q1 : F, + B , q2 : F2 + A / B epimorphisms where F,, F2 are free. Then there is an epimorphism q : F, @ F2 + A , such that 9 I Fl = q1 and qg B = q 2 g for g E F 2 . Let A , ( n = 0, -t 1, k 2 , . . .) be arbitrary groups. Prove the existence of free groups F'") and a sequence

+

11.

...

~

F("- 1 )

an- I

,F(n)2+ F(n+ 1 ) . . . --$

such that N , , U , - ~ = 0 and Ker cc,/Im 12. Prove (14.7) in detail.

E A,

for every n.

15. FINITELY GENERATED GROUPS

We shall find the following lemma useful

Lemma 15.1. Let A be a p-group and assume that A contains an element g

of maximal order pk. Then ( 9 ) is a direct summand of A .

Let B be a (g)-high subgroup of A . In order to prove that A = (g) @ B , we recall (9.9) and show that pa = mg b ( a E A , b E B , and m E Z) implies p I m. By the maximality of the order of g, pk-'rng p k - ' b = pk a = 0. Hence pk-'rng = 0, and in must be divisible by p . 0

+

+

Now we can turn t o the first proper structure theorem in the history of group theory.

Theorem 15.2 (Frobenius and Stickelberger [l]). A jinite group is the direct sum of ajinite number of cyclic groups of prime power orders. Because of (8.4), we can restrict ourselves to p-groups. If A # 0 is a finite p-group, then we select an element a E A of a maximal order pk. By the preceding lemma, A = ( a ) 0B for some B < A . Since B is of smaller order than A , a trivial induction completes the p r o o f . 0 Having discovered the structure of finite groups, we proceed to finitely generated groups. We prove two preliminary lemmas.

78

111. DIRECT SUMS OF

CYCLIC GROUPS

Lemma 15.3 (Rado [l]). If A = ( a , , . . . , a,) and the integers n,, . . . , nk satisfy (n,, . . . , nk) = I , then there exist b, , . . . , bk in A such that

A=(b,;..,b,)

where b, = n , a , + . . . + n , a , .

If n = Inl[ + . . * + In, I = 1, then b, = +a, for some i, and the assertion is trivial. Therefore, let n > I and apply induction on n. Now at least two of the n, are different from 0, say, Inl] 2 In,l > 0. We have either In, + n,l < Inl] or In, - n21 < In,l, hence In, n,l ln21 . . . Ink[< n for one of the two signs. By (n, f n, , n, , . . . , n,) = 1 and by induction hypothesis,

+ +

A

=

+

+

(a,, a 2 , . . ., ak>= (al, a2 T a,, a 3 , ..., a,)

with b, = (n, f n2)al

=

(b,, b,, . . ., b k )

+ n2(a2f a,) + n3 a3 + . . . + nk ak = nlal + . . . + nkak.O

It is convenient to say that { a , } i Eis, a basis of A if A is the direct sum of the cyclic groups ( a i ) ( i E I ) . Lemma 15.4. Let H # 0 be a subgroup of a free group F of rank n. Then F has a basis a,, . . ' , a, and H has a basis b, , ' . . , b, such that

bi=miai

(i= l;..,n)

where the m i are nonnegative integers satisfying m i - , Imi

( i = 2, ..., n).

We start with a basis c,, . . . , c, of F with the following minimal property: H has an element b, = k,c, + ... + k,c, with a minimal positive coefficient k , . In other words, for another basis of F or for another arrangement of c,, . . . , c,, or for other elements of H , the first positive coefficient is never less than k,. Then k , I k i ( i = 1, . . . , n), for if k , = qi k , + ri (0 5 ri < k,), then we can write b, = k,a, + r, c2 + . . . + r, c, where a, = c, + q, c, . . . + q, c,, c 2 , . . ' , c, is a basis of F, and by the choice of c,, c, we must have r2 = . . . = r, = 0. The same argument shows that if b = slcl + . . . + s, c, E H , then s, = qk, for some integer q. Hence b - qb, E ( c 2 ) 0.. . @ (c,). We infer that F = ( a , ) @ (c,) @ . . . @ (c,) and H = ( b , ) @ H , where b, = k,a, and H , 5 ( c 2 ) @ ... @ (c,). A simple induction on n shows the existence of a basis a,, ..., a,, of F a n d one b,, ' . . , b, of H satisfying bi = m i a i . In order to establish the last statement, we show that m, I m,. Write m2 = qm, + r (0 5 r < m,) and replace the basis element a, by a = a, + qa, . In terms of the new basis a, a,, * . . , a,,, the element b, + b, E H has the form b, + b, = mtal + (qm, r)a2 = m,a + ra, . By the minimality of m, = k , , we necessarily have r = 0 . 0

+

. . a ,

+

The main result on finitely generated groups is the following famous theorem.

15.

79

FINITELY GENERATED GROUPS

Theorem 15.5. The follolling conditions on a group A are equivalent :

(i) A isjinitely generated; (ii) A is the direct sum of a finite number of cyclic groups; (iii) the subgroups of A satiJfy the maximum condition. Condition (i) implies (ii). Since this is the most essential portion, we give two independent proofs for it. The first is based on (15.3). Assume A is finitely generated and every generating system of A contains at least k elements. Choose a system of k generators a , , . . . , ak such that, say, a, is of minimal order, i.e., n o other set of k generators contains an element of smaller order. By the choice of k , o(al) > I . As a basis of induction, we may assume that B = ( a , , . . ., a k ) is a direct sum of cyclic groups. Consequently, it suffices to show that A = ( a , ) @ B , which will follow if we can show that ( a , ) n B = 0. By way of contradiction, let us assume mlal = m 2 a 2 + . . . + mkak # 0 with 0 < m, < o(al). Set (m,, ..., mk) = m and write m i= mn,. Then (n,, ..., nk) = 1, and from (15.3) we infer ( a , , ..., a h ) = ( b , , ..., b k ) with b, = - nlal + n, a, + . . . + n k ah. Here mb, = 0 , o(b,) < o ( a l ) contradicts the choice of a , . The second proof makes use of (1 5.4). If A is generated by n elements, then by (14.3) we may write A z F / H , where F is a free group of rank n. If we choose bases of F and H as in (15.4), then we obtain

Thus A is a direct sum of cyclic groups: if m i= 0, then the ith summand is an infinite cyclic group, otherwise it is cyclic of order m i . Condition (ii) implies (iii). Let A = (a,) 0 . . .0 (a,,). If n = I , then A is cyclic, and every nonzero subgroup of A is of finite index; thus the subgroups satisfy the maximum condition. As a basis of induction, assume that in B = ( a l ) 0. . . @ (a,-,), the subgroups satisfy the maximum condition. If C, S C , 5 ... is a n ascending chain of subgroups in A , then B n C, 5 B n C, 5 . . . is one in B , and so, from some index k on, all B n C, are equal (to B n Ck). For m > k we have Cll,/(Bn Ck> = C,,/(B n C,)

g

( B + C,,)/B 5 A / B ,

where A / B z (a,,) is cyclic. Consequently, in the ascending chain C k / ( Bn C k ) 5 Ck+l/(B n ck)5 . . - , from a certain index k + I on, all groups are equal, i.e., Ck+l = Ck+l+, = ... . Finally, (iii) implies (i). Let the subgroups of A satisfy the maximum condition. The set of all finitely generated subgroups of A is not empty, hence A has a maximal finitely generated subgroup G. For every a E A , together with G, (G, a ) is also finitely generated; hence G = (G, a ) and G = A . 0

80

111.

DIRECT SUMS OF CYCLIC GROUPS

We notice that (1 5.5) implies that subgroups offinitely generatedgroups are again finitely generated. It is natural to raise the question as to the uniqueness of the decomposition of a finitely generated group into the direct sum of cyclic groups. If the order of a finite cyclic group is divisible by at least two primes, then the group can be decomposed into a direct sum of groups of prime power orders, so-in order to get some kind of uniqueness-we may, and shall, restrict ourselves to direct decompositions into cyclic groups of infinite or prime power orders. Then uniqueness [up to isomorphism] will be a simple consequence of the subsequent theorem that gives a description of the subgroups of a finitely generated group.

Theorem 15.6. Let B be a subgroup of thefinitely generatedgroup A . Then : (i) the number s of infinite cyclic groups in a decomposition of B into direct sums of cyclic groups does not exceed the same number r for A ; (ii) if p r l 2 .. . 2 prk > 1 are, f o r a fixed p , the orders of the cyclic p groups in a direct decomposition of A into cyclic groups of infinite and prime power orders, and if p" 2 * 2 p"" > 1 have the same meaning for B, then k 2 m and ri 2 si f o r i = 1, ..., m ; (iii) if C is a direct sum of cyclic groups such that the orders in a decomposition of C satisfy (i) and (ii), then C is isomorphic to a subgroup of A . Because of T(B) = B n T(A), we have B/T(B) z ( B + T(A))/T(A) A/T(A), where the quotient groups are direct sums of s and r infinite cyclic groups. Thus, to verify (i), we may assume A torsion-free. Then A , B are free groups of rank r, s, and, by (15.4), B has a basis consisting of, at most, as many elements as some basis of A . Hence (14.1) yields s 5 r. In order to prove (ii), let us begin by observing that lA[p]l = p k and IB[p]1 = p m , whence k 2 m is obvious. By way of contradiction, assume rl 2 sl, ..., 2 si-l, but ri < si for some i. Then thep-component of p r i A will be the direct sum of cyclic groups of orders prl-", ..., pri-'-" , while priB that of cyclic groups of orders psi-", . . . ,psi-", . . ., thus I(priA)[p]l= pi-' < p i 5 I(priB)[p]I would be a contradiction. Assertion (iii) is evident. 0 If B = A , (15.6) yields the unicity of orders of the cyclic groups in decampositions of A into direct sums of cyclic groups of orders infinity or powers of primes. [It should, however, be emphasized, that these cyclic groups are not uniquely determined subsets of A . ] These orders are called the invariants of A ; for instance, the invariants of A = 2 0 Z @ Z ( p ) @ Z ( p 2 ) @ Z(qz)are co, c o , p , p 2 ,q 2 . In this case, we also say: A is of type (co,00, p , p 2 , q2). We have thus ordered to every finitely generated group A a finite system consisting of symbols co and prime powers, the invariants of A . Not only are

15.

FINITELY GENERATED GROUPS

81

these uniquely determined by the group, but they determine the group, that is to say, two finitely generated groups are necessarily isomorphic if they have the same system of invariants-this fact is'usually expressed by saying that we have a complete system of inrariants. Moreover, these invariants are independent in the sense that to every finite system of symbols 00 and prime powers, there exists a finitely generated group whose system of invariants coincides with the given system. The description of a class of groups by means of invariants is one of the major aims of group theory. In most cases, the invariants are natural integers, cardinal or ordinal numbers, but they could be matrices with integral entries, etc. The invariants must be easily describable quantities and-as their name indicates-must be uniquely determined by the groups; they are satisfactory if the isomorphy of groups can easily be recognized from their invariants. The systems of invariants for finitely generated groups are most satisfactory, and they have served as a classical model for structure theorems of algebra. To be more precise, a structure theorem for a class of algebraic systems consisis in finding a complete [and possibly independent] system of invariants along with a method of reconstructing, from the invariants, the algebraic system we started with. In the case of finitely generated groups, this construction consists in forming the direct sum of cyclic groups with the given invariants as orders.

(15.5) can be generalized in various ways to modules. For our purposes, it suffices to consider modules over the p-adic integers : Theorem 15.7. For modules over thep-adic integers, (15.3)-( 15.6) hold true. The proofs given above carry over mutatis mutandis. 0

EXERCISES 1. (a) A group is finite if it has but a finite number of subgroups. (b) A group is finite if its subgroups satisfy both the maximum and the minimum condition. 2. A finite group A is cyclic if and only if for every prime p , IA[p]I 5 p . 3. The number of nonisomorphic groups of order m = p;' . . .p;;"is equal to P ( r l ) .. . P(r,), where P ( r ) denotes the number of partitions of r into positive integers. 4. Let C(m) denote the multiplicative group of residue classes prime to m , modulo the integer m = p : ' . . .p;;". (a) C(m) is the direct product of the C(p;'),i = I , . * * , k ; (b) C(pk)is cyclic if p is an odd prime; (c) C(4) is cyclic, while C(2"), n >= 3, is of type (2, 2"-'); (d) every finite group is isomorphic to a subgroup of some C(m). 5. If m divides the order of A , then A has subgroups and quotient groups of order m. 6. Prove assertion (i) of (15.6) with B a quotient group, rather than a subgroup of A . Is (ii) true for quotient groups?

82

111.

DIRECT SUMS OF CYCLIC GROUPS

7. If A , B are finite groups such that, for every integer m,they contain the same number of elements of order m , then A z B. 8. Given two finite groups A , B , there exists a group C such that both A and B have direct summands isomorphic to C, and every group isomorphic to direct summands of both A and B is isomorphic to a direct summand of C. 9. Let A have the invariants pk,. . . ,pk (n times). A subgroup B of A is a direct summand of A if and only if every invariant of B is pk. 10. (Szele [l]) Let B be a subgroup of the finite p-group A . B is a direct summand of A exactly if there are direct decompositions A = A , 0 .. . 0 A , , B = B, 0 - .. 0 B, such that Bi 5 A i(i = 1, . . -,rn) and A i , Bi have equal invariants pi. 1 1 . A generating system a,, . . *, ak of a finite group A is a basis of A if and only if the product o ( q ) . .. o(ak)assumes its minimum [which must be 1,411. 12. Establish the existence of a basis g,, . . . , gr of a finite group A satisfying the divisibility conditions o(gl) lo(g2)l * * I o(g,), and prove that for all such bases of A , the orders o(g,), . . ., o(g,) are the same. 13. (Birkhoff [l]) Let A = (al) 0 ... 0 (a,,), where o(ai) = p r * with r l z . . - z r , ,Call . b = s , a , + . . . + s , , a , and c = t l a l + - . . + t , , u , , orthogonal if

C si tiprl-ri= 0 mod p". i

Show that the elements of A orthogonal to all the elements of a subgroup B form a subgroup C such that IBI * ICI = IAl. 14. (a) Subgroups and quotient groups of elementary groups are elementary. (b) An elementary group E of order p' contains (p' - l)(p'-l - 1) ... (p'-'+' - 1)/(p - l)(p2 - 1) ... (p'

15. 16.

17. 18.

-

1)

subgroups of order p' ( t 5 r ) . (c) E has (p' - l)(p'-' - 1) . . . ( p - l)/(p - 1)' composition series. (d) Determine the number of different bases of E. (G. Frobenius) In a finite p-group A , the number of subgroups of a fixed order [dividing lAl] is F 1 modp. (a) The sum of all the elements of a finite group A is 0, unless A contains a single element a of order 2, in which case the sum is a. (b) Derive Wilson's congruence ( p - I)! = - 1 m o d p from (a). Given the invariants co, . . . ,pk, ., q1 of a finitely generated group A , determine the minimal number of elements in a generating system of A . (Kaplansky [2]) If A is an infinite group all of whose nonzero subgroups are isomorphic to A , then A E Z .

16.

83

LINEAR INDEPENDENCEAND RANK

19. (Fedorov [l]) An infinite group is cyclic exactly if all of its nonzero subgroups are of finite index. 20. If the subgroups of both A / B and B satisfy the maximum condition, then so do the subgroups of A . 21. In a finitely generated group, every generating system contains a finite set of generators. 22. If F = (x,) 0.. . @(x,,) is a free group of rank n, G = (yl) 0 ... @ ( y , ) a subgroup of the same rank where yi=

23. 24.

25. 26.

,x

J =

1

rijxj

( i = 1,

..a,

n),

then the order of FIG is equal to the absolute value of the determinant of the matrix [Irij(l. The cardinal number of the set of pairwise nonisomorphic finitely generated [finite] groups is countable. (Cohn [l], Honda [2], E. A. Walker [l]) Let A = B G = C 0H where B and C are isomorphic finitely generated groups. Then G E H . [Hint: assume G Z H ; if B z Z , G = Z O ( G n H ) ; if B = ( b ) is of order pk and pk-’b 4 H , then A = B OH, while if pk-’b E H , C = ( c ) , p k - ’ c E G, then A = ( b + c) @ G = (b + c) OH.] (E. A. Walker [l]) Prove the isomorphism of direct decompositions of a finitely generated group by making use of Ex. 24. (a) If A , B are finitely generated groups each of which is isomorphic to a subgroup of the other, then A z B. (b) The same with “subgroup” replaced by “quotient group.” 16. LINEAR INDEPENDENCE AND RANK

For the selection of a basis in a direct sum of cyclic groups, it is convenient to have the concept of linear independence at our disposal. This used to be defined in abelian groups in two inequivalent ways; here we shall adapt what is not trivial for torsion groups and is therefore more useful for our purposes (see Szele [2]). A system {a,, . . ., a,} of nonzero elements of a group A is called linearly independent, or briefly independent, if (1)

n,a,

+ . . . + nk a, = 0

(ni E Z)

implies

nlal

=

. . . = n, a,

= 0.

More explicitly, this means n, = 0 if o(a,) = co and o(a,) I n, if o(a,)is finite. A system of elementsisdependent if it is not independent. [Notice that (unlike for

111.

84

DIRECT SUMS OF CYCLIC GROUPS

vector spaces) it is, in general, not true that one of the elements in a dependent system can be written as a linear combination of the others.] An infinite system L = { a , } i s lof elements of A is called independent, if every finite subsystem of L is independent. Thus independence is, by definition, a property of finite character. An independent system cannot contain equal elements, hence it is a set.

,

Lemma 16.1. A system L = {a,} I is independent if and only i f the subgroup generated by L is the direct sum of the cyclic groups (a,), i E I. If L is independent, then for each i E I, the intersection of ( a , ) with the subgroup generated by all a j with a j E L ,j # i, is necessarily 0 ; hence ( L ) is the direct sum of the groups ( a , ) , iE I. Conversely, if ( L ) = O i E(a,), I then 0 can be written in the form nlai, + ... + nkai, = 0 with different i,, * , ik from I only in the trivial way: nlail = . . . = nka,, = 0. This shows the independence of L . 0 g E A is said to depend on a subset L of A if there is a dependence relation

+- +

0 # ng = nlal . . nka, (2) for some ai E L and integers n, n, . A subset K of A depends on L if every g E K depends on L. If both K depends on L and L on K , then K and L are said to be equivalent, An independent system M of A is maximal if there is no independent system in A containing M properly. Thus, if g E A , g # 0, then { M , g} is no longer independent, and g depends on M . It is clear that any two maximal independent systems in a group A are equivalent. By Zorn’s lemma, every independent system in A can be extended to a maximal one. Moreover, if the initial independent system contained only elements of infinite and prime power orders, then the same can be assumed of the maximal one. In fact, every element a of finite order in an independent system can be replaced, without violating independence, by an arbitrary multiple ma # 0, and so by one of prime power order, too. A subgroup E of a group A is called essential if E n B # 0 whenever B is a nonzero subgroup of A. [Clearly, it suffices to state this for cyclic B.] In this case, A is said to be an essential extension of E. For instance, the socle is an essential subgroup of a torsion group. We have the next simple result. Lemma 16.2. An independent system M of A is maximal ifand only i f ( M ) is an essential subgroup of A. Every maximal independent system in an essential subgroup of A is maximal independent in A . Sinceg E A depends on M if and only if ( M ) n ( 9 ) # 0, the first part of the lemma is evident. If E is essential in A and M is a maximal independent system in E, then let g E A be arbitrary. We have a nonzero h E E n ( g ) , h = mg. Since h depends on M , so does g, t o o . 0

16.

85

LINEAR INDEPENDENCE AND RANK

By the rank r(A) of a group A is meant the cardinal number of a maximal independent system containing only elements of infinite and prime power orders. If we restrict ourselves t o elements of infinite order in A , i.e., we select an independent system which contains elements of order cc only and which is maximal with respect to this property, then the cardinality of this system is called the torsion-free rank r,(A) of A . The p-rank rp(A) of A is defined analogously, by using elements whose orders are powers of a fixed prime p rather than elements of infinite order. It is clear from these definitions that the following equality holds true for every group A : (3)

P

with p running over all primes. Evidently, r ( A ) = 0 amounts to A

= 0.

The main theorem on ranks reads as follows. Theorem 16.3. The ranks r(A), r,(A), r,(A) of a group A are invariants of A . By (3), it suffices to verify the invariance of r,(A) and rp(A). The proof of invariance of ro is reduced to torsion-free groups if we can show that r,(A) = r ( A / T ) ,where T is the torsion part of A . If a l , . . ., ak E A are independent and of infinite order, and if the cosets a: = a, + T satisfy nlaT + . . * + nk at = 0*, then nlal + . . . + nk ak = b E T. Multiplication by in = o(b) gives rnnlal + . . . + / m k ak = 0, whence rnn, = 0, n, = 0 for i = 1, . . . , k , i.e., {a:, . . . , a,*} is an independent system in A / T . Conversely, if this is a n independent system in A/T, then nlal ... nkak = 0 implies nlaT + ... + nka? = o*, whence ni = 0 for every i. This proves r,(A) = r ( A / T ) as claimed. In order to verify the invariance of r ( A ) for a torsion-free A , choose a maximal independent system L = { a i } i EinI A. For g E A , g # 0, we have then ng = n l a i , + . . . + nkaik # 0. If we associate with g a system {il, .. ., ik ; n, n,, .. ., n k } consisting of a finite number of indices and nonzero integers, then n o other g' E A is associated with the same system, since if both g and g' define the same system, then n(g' - y) = 0 and g' = g. Consequently, the cardinality of A does not exceed that of all systems {il, . . . , ik ; n, n,, . . . , nk} which implies IAl 5 111 No = r ( A ) ' N o , Since trivially r ( A ) 5 IAl, we see that r ( A ) = IAl whenever r(A) is infinitc. I f A has a finite maximal independent system {al, ..., ak}, and if { b , , ..., b,} is any independent system in A , then the second system depends on the first; hence for some integer rn > 0, rnb, E ( a l , ..., ak) for i = 1, ..., 1. Thus (n7b1) 0 . . . @ (rnb,) is a subgroup of ( a l ) @ . . . @ ( a k ) , and so, by ( I 5.6), the inequality I 5 k holds. We infer that every independent system in A contains at most k elements, and therefore every maximal independent system in A consists of the same number of elements. Thus r,(A) is a n invariant for every A .

+ +

86

111.

DIRECT SUMS OF CYCLIC GROUPS

Turning to the ranks rp(A), it is clear that r,(A) = r(Tp),where T, is the p-component of A. Hence we need only prove the invariance of r(A) for p groups A . If A is a p-group and S(A) = A [ p ] is its socle, then r(A) = r(S(A)). In fact, a system {al, . . . , ak} in A is independent if and only if {pml-lal,.. . , pmk-'uk}is independent where mi = e(ui). Therefore, only the uniqueness of r(S(A))is to be verified. As S(A) is, in the natural way, a vector space over the field F, of p elements, and independence in the sense above coincides with the vector space independence, we see that r(S(A)) is just the dimension of the vector space S(A), and hence it is unique.0 From the last part of the proof we get, in view of (16.1): Corollary 16.4. In an elementary p-group, every maximal independent system is a basis.0 It is clear how to define independence and rank for modules over thep-adic integers. Notice that in this case q-ranks for primes q # p are trivial, so one may confine himself to torsion-free and p-ranks for one p . As in (16.3), one concludes that they are invariants of the module.

EXERCISES 1. The subgroups Bi ( i E I) of A generate their direct sum in A if and only

2.

3.

4.

5. 6.

if every set L = { b i } i , rwith one bi (#O) from each.Bi, is independent. If K is an independent and L a maximal independent system in A such that L contains only elements of infinite and prime power orders, then A has a maximal independent system containing K and contained in K u L. Let B be a subgroup of A . Show that: (a) r(B) S 4 4 ; (b) 4 4 ) 5 0) + r(A/B); (c) r(A) < r ( A / B ) can happen; ( 4 r o ( 4 = ro@) + ro(A/B). If Bi (i E I) are subgroups of A such that A = B i , then r(A) 5 r(Bi) where equality holds if the sum is direct. A is locally cyclic if and only if ro(A) + maxp r,(A) 5 1. (a) A contains no subgroup which is the direct sum of two of its proper subgroups if and only if r ( A ) 5 1. (b) r ( A ) 5 1 if and only if A is isomorphic to a subgroup of Q or some

zi

Z(P").

cisl

7. If B,, . . . , B, are subgroups of A such that A / B i are of finite rank, then A/(Bl r\ ... n B,,) is again of finite rank. 8. A group of infinite rank lit contains exactly 2'" different subgroups.

17.

87

DIRECT SUMS OF CYCLIC p-GROUPS

9. If E is an essential subgroup of A and B 5 A , then E n B is essential in B. 10. A subgroup B of A is essential if and only if S(A) 5 B and A / B is

torsion. 1 1 . The essential subgroups of a group A form a dual ideal in L(A). This is principal if and only if A is torsion. 12. If B is a subgroup of A , then there exists a subgroup of A which is maximal with respect to the property of being an essential extension of B in A . 13. (a) For a p-group, the rank is the same if it is regarded as a Z-module or as a Q,*-module. (b) The group J p is as an abelian group of rank 2'O and as a Q:-module of rank 1. 17. DIRECT SUMS OF CYCLIC p-GROUPS

Let A be a direct sum of cyclic p-groups. Then, evidently, A contains no elements # O of infinite height. We shall see by an example below that the absence of elements of infinite height does not imply that a p-group is a direct sum of cyclic groups. It is of importance to have criteria under which a given p-group is a direct sum of cyclic groups.

Theorem 17.1 (Kulikov [I]). A p-group A is a direct sum of cycZic groups i f and only if A is the union of an ascending chain of subgroups (1)

A , 5 A2 5 * . . 5 A , 5 . . . )

u m

A,

n= 1

= A,

such that the heights of elements f O of A , are less than aJinite bound k,. If A is a direct sum of cyclic groups, then in a decomposition, collect the cyclic direct summands of the same order p", for every n, and denote their direct sum by B,. Clearly, A, = B, 0 . . . 0 B, satisfy the conditions with k, = n. For the proof of the sufficiency, assume that the subgroups A , of A satisfy the hypotheses. Since we may adjoin 0's to the beginning of (1) and repeat some terms A,, [a finite number of times], it is clear that there is no loss of generality in assuming k, = n, that is, A , n p"A = 0 for every n. We consider all chains

c15 c25 . . ' 5 c, 5 ...

of subgroups C, of A , such that (i) A , 5 C,

and

(ii) C, n p " A = O

for all n , and define the chain of the C, less than or equal to the chain of the B, if C, 5 B, for every n. Then the set of all chains with (i) and (ii) is inductive,

111.

88

DIRECT SUMS OF CYCLIC GROUPS

and Zorn's lemma applies to conclude the existence of a chain GI5 G, 5 - . . 5 G, 5 ... satisfying (i) and (ii) [with C, replaced by G,] which is maximal in our present sense. Evidently, UG, = A. For every n, we select a maximal independent set L, of elements in the subgroup G,[p] n p " - I A , and define L as the union of all the L, ( n = 1 , 2 , . .). For every ciE L with mi = h(ci) we choose an a , E A satisfying pmiai= c i . We claim that A' = (..., a i , ...) = @ ( a i ) coincides with A . The first step in the proof of this is to show that ( L ) = A [ p ] . Owing to (16.4), ( L , ) = G,[p] n p " - l A . Since the elements #O of (L,) are all of height n - 1, we see that the (L,) generate their direct sum, ( L ) = @,(L,). Assume, as a basis of induction, that every a E A [ p ] contained in G, belongs to ( L ) [which obviously holds for r = I], and let b E G,+,[p]\G,. Then b $ G , implies (G,,b) n p ' A # O ; let O # g + k b = c ~ p ' A , w h e r e g E G,, and k = 1 may be assumed [for otherwise we multiply by k' with kk' = 1 mod p ] . Here c E G,+l and h(c) 2 r ; thus, by (ii), o(c) = p and h(c) = r. We infer c E (L,+,). Furthermore, pg = p c - pb = 0 and g E G,, together with theinduction hypothesis, imply g E ( L ) . Hence b = c - g E ( L ) , establishing ( L ) = A [ p ] . Assume we have proved that every a E A of order S p " belongs to A', and let b E A be of order p n + l with n 2 1. As is proved in the preceding paragraph, p"b E ( L ) , and so p"b = mlcl + ... + mk ck with some ci E L . Let c1;..,cj b e o f h e i g h t 2 n and C ~ + ~ , . . . ,ofheight C~ s n - 1 . I f w e write mici = p n m i a i for i = 1, . . . , , j , then we have +

p"(b - m ; a l

-

... - m'.a J J.) = m ,.+ l c j + 1 + ... + nzkck E G,-l.

Condition (ii) implies b - m;al - ... - m j a j is of order Sp", thus in A', and consequently, b E A'.O As corollaries we obtain the following two important results.

Theorem 17.2 (Prufer [2], Baer [l]). A bounded group is a direct sum of cyclic groups. If A is a bounded group, then its p-components are again bounded. If we choose for all the members of (1) the p-component of A , then we conclude from (17.1) that the p-components of A are direct sums of cyclic g r o u p s . 0 Theorem 17.3 (Priifer [I]). A countable p-group is a direct sum of cyclic groups, $and only if it contains no elements # 0 of injinite height. Only the "if" part needs a verification. Let A be a countable p-group without elements of infinite height and {a,, . . . , a,, . . .} a generating system for A . Now A is the union of its finite subgroups A , = ( a l , ..., a,) (n = 1, 2, . . .), where the heights of the elements are trivially bounded. (17.1) completes the pro0f.O

17.

DIRECT SUMS OF CYCLIC p-CROUPS

89

The fol1owir.g example, given by Kurosh [2], shows that countability is an essential hypotixsis in (17.3). Let A denote the torsioi: part of the direct product of the cyclic groups Z ( p ) , Z ( p z ) ,. . . , Z(p"),. . . . Then A is a p-group of the power of the continuum, without elements of infinite height. Contrary to our assertion, assume A = @ ,, A , where A, is a direct sum of cyclic groups of the same order p". The socles S,, = Oy==, A i [ p ] form, hith increasing n, a descending chain, where S, consists exactly of those eleinents of S , = A [ p ] which are of height z n - I . Clearly, a = ( b l , b,,

. . . , b,, , . . .>E fl Z(p"), n

b,

E

Z(p"),

is of height >=n - 1 only if b, = . . . = h,,-, = 0. This shows that each quotient we group S,,/S,,+,( n = 1, 2, . . .) is of order p. l n view of A , [ p ] E SJS,,,, infer that t h e 2 , are finite, and hencc A is countable. This contradiction proves that A is not a direct sum of cyclic groups. If A is a direct sum of cyclic p-groups, then it may have a number of such direct decompositions. But there is a unicity as far as the orders of the components are concerned :

Theorem 17.4. Any two decon1positioii.r of a group into direct sums of cyclic groups of infinite and prime porr'er orders are isomorphic. Let S, be the subgroup of S = A [ p ] consisting of elements of height 2 n - 1. If A is a direct sum of cyclic p-groups, then S, is the direct sum of the socles of all direct summands of orders >,p". Consequently, S,,/S,+, is isomorphic to the direct sum of the socles of direct summands of order p" in a direct decomposition of A . We infer that the number of direct summands of order p" in a direct decomposition of A with cyclic summands is equal to the rank of Sn/S,,+l. The S, were defined without any reference to direct decompositions, hence thecardinal number titpn of the set of direct summands of order p" in any decomposition of A into direct sums of cyclic p-groups is the same. Since the number i t l o of direct summands Z in a direct decomposition of a group A into cyclic groups is equal to r,(A), we conclude from (16.3) that i n o , too, is uniquely determined by A . 0 The preceding theorem states that for direct sums of cyclic groups Z and Z(p"), the cardinal numbers l i t o and i l l p n are invariants. They constitute a complete and independent systcni of invariants for direct sums of cyclic groups.

EXERCISES 1.

(a) A group A is elementary if and only if every subgroup of A is a direct summand.

90

2. 3.

4. 5.

6.

7.

8.

9*.

10.

11.

12.

111.

DIRECT SUMS OF CYCLIC GROUPS

(b) A is elementary exactly if it is a torsion group whose Frattini subgroup vanishes. (c) A is elementary if and only if A is the only essential subgroup in A . A [ n ] is always a direct sum of cyclic groups. If A is a direct sum of finite cyclic groups, then A / m A E A [ m ] for any integer m > 0. Let B be the direct product of in copies of Z ( p k )with fixed pk, and A the direct product of n copies of B. Then A is a direct sum of groups Z(pk); determine the cardinality of the set of the components. Describe the groups in which: (a) every maximal independent set is a basis; (b) every generating system contains a basis. Prove (17.2) for p-groups as follows: select an independent system of elements of maximal order p”, extend it successively by elements of orders p k - ’ , . . . , p , such that in each step the independent system is as large as possible, and show that the arising set is a basis. (Szele [3]) Let A be a bounded p-group and p r the maximal order for elements in A . Then a E (pr/o(a))Afor every a E A if and only if A is a direct sum of cyclic groups of the same order pr. (Dlab [4]) (a) If A is a bounded p-group and S = is a set of elements in A , such that ai+ p A (i E I ) generate A / p A , then S generates A . (b) Every generating system of a bounded p-group contains a minimal generating system. (c)* For unbounded p-groups, (b) fails. (Dieudonnk [2]) Generalize (17.1) as follows: if G is ap-group containing a subgroup A , such that G / A is a direct sum of cyclic groups and (1) holds where the heights of elements of A , , taken in G, are bounded, then G is a direct sum of cyclic groups. Prove (17.3) as follows: start with a maximal independent system in the socle, cl, . . ., c,, ..., and consider all b with p k b = mlcl + . . . + m,c, with m, c, # 0 for a fixed n. Choose a b, say b,, with maximal order, for every n, and prove that b,, . . . , 6, , . . . is a basis. Let A , B be direct sums of cyclic groups. (a) A @ A z B @ B implies A 2 B. (b) A @ ... 0 A .. B @ ... @ B @ ... does not imply A E B, not even if A , B are finitely generated. Let A be the direct sum of a countable set of cyclic groups of order p 2 and B = A @ Z ( p ) . Show that A , B are not isomorphic, but bothqua abstract groups-have the same set of subgroups and the same set of quotient groups.

18.

SUBGROUPS OF DIRECT SUMS OF CYCLIC GROUPS

91

13. Let A = A, @ ... @ A, where A i is the direct sum of mi copies of Z ( p i ) ,i = 1, . . . , k. Describe the structures of subgroups and quotient groups of A. 14. (a) Let G = Z(pk)>.Prove that every countable p-group is an epimorphic image of G. (b) Every p-group of cardinality tit is an epimorphic image of a direct sum of m copies of G in (a). 18. SUBGROUPS OF DIRECT SUMS OF CYCLIC GROUPS

We have seen that subgroups of free groups are free, and from (17.3) it follows trivially that subgroups of countable direct sums of cyclic p-groups are again direct sums of cyclic groups. These are special cases of the following general result.

Theorem 18.1 (Kulikov [2]). Subgroups of direct sums of cyclic groups are again direct sums of cyclic groups. First, let A be a direct sum of cyclic p-groups, and B 5 A . Then A is the union of an ascending chain A , 5 . . . 5 A, 5 . . * of its subgroups, where the heights of elements of A,, are bounded, say, sk,, . Evidently, B is the union of the ascending chain B, 5 ... 5 B, 5 . . .

with

B, = B n A,,,

and the heights of elements of B,, , relative to B , do not exceed k, . By virtue of (17.1), B is a direct sum of cyclic groups. Thus our theorem holds for torsion groups. Let A be an arbitrary direct sum of cyclic groups. If T is its torsion part, then B n T is the torsion part of the subgroup B of A. Now B/(B n T ) r ( B + T ) / T j A/T, where A/T is a free group. By (14.5), B/(B n T ) is free, and so (1 4.4) implies B = ( B n T ) @ C for some free subgroup C of B. By what has been shown in the preceding paragraph, B n T is a direct sum of cyclic p-groups; thus B is a direct sum of cyclic groups.0

Corollary 18.2 (Kulikov [ 2 ] ) . Any two direct decompositions of a group which is a direct sum of cyclic groups have isomorphic refinements. In view of (18.1), each direct summand is a direct sum of cyclic groups. If we replace each direct summand by a direct sum of cyclic groups of order co or prime power, we arrive at refinements which are isomorphic, as is shown by (17.4).0 Under certain circumstances we can conclude that A is a direct sum of cyclic groups if a subgroup of A is a direct sum of cyclic groups.

111.

92

DIRECT SUMS OF CYCLIC GROUPS

Proposition 18.3 (Fuchs [l], Mostowski and Sqsiada [l]). If B is a subgroup of A such that B is a direct sum of cyclic groups and AIB is bounded, then A is a direct sum of cyclic groups. By hypothesis we have nA 5 B 5 A for some integer n. From (18.1) we conclude that n A is a direct sum of cyclic groups, and so it is enough to show that A is a direct sum of cyclic groups if p A is for some prime p. Let p A = @ I ( b , ) , where o(b,) is infinite or a power of prime, for every i E I. Choose a, E A such that pa, = b i for i E I, with the proviso that a, E ( b i ) if o(bi) is prime t o p . Extend the independent system {a,} I by a set { c j } to obtain a maximal independent system { a , , c~},,~ in A . We claim A is the direct sum of the ( a , ) and ( c j ) . We need only show that every a E A lies in this direct sum. Clearly, p a = n,b, + . . . n k b, = p(n,a, + . . . + n k a k ) , and so a’ = a - nlal - .. . - nkak is in the socle of A . Since a’ depends on {a,, c ~ } ,and , ~ therefore is expressible as a linear combination of the a i , c j , the same holds for a . 0

,

,

+

From (18.3) we can easily derive: Corollary 18.4 (Kulikov [ 2 ] ) .Every group A is the union of an ascending chain A , 5 . . . 5 A,, 5 . . . of subgroups where every A, is a direct sum of cyclic groups. Define A , as generated by a maximal independent system of A . If, for some n, A,-, is defined, then let A , = n-’A,-, = { a E A Ina E A,-,}. From the choice of A , it follows that the union U A , for all n is equal to A , while from (18.3) we infer, successively, that A, is a direct sum of cyclic groups.0 The results of this section extend immediately to p-adic modules.

EXERCISES

fl

A with an 1. Let A be a p-group. The torsion part of a direct product infinite number of components is a direct sum of cyclic groups if and only if A is bounded. 2. If both B and A / B are direct sums of cyclic groups and if i n AIB the elements of finite order are of bounded order, then A is a direct sum of cyclic groups. 3. (Szele [ 5 ] ) Let A

=

(a,, a,, *.., a,, ...; a,

= mlal =

... = mnan= .-.)

where m, are positive integers. A is a direct sum of cyclic groups if and only if the system {w,}, is bounded. 4. Let A = B + C where both B and C are direct sums of cyclic groups. Then A need not be a direct sum of cyclic groups.

19.

COUNTABLE FREE GROUPS

93

5. Relate the invariants of a direct sum of cyclic groups to those of its subgroups. 6 . Prove (18.1) for modules over the p-adic integers. 7. An equational class or variety of groups is a class of groups that is closed under formation of subgroups, quotient groups, and direct products. (a) Prove that the following is the list of all equational classes of abelian groups: (i) all abelian groups; (ii) for any positive integer n, all abelian groups annihilated by n. (b) Show that the smallest variety containing Z [or Z(n)] is the one given in (i) [in (ii)]. 19. COUNTABLE FREE GROUPS

Kulikov’s theorem (17.1) yields a handy necessary and sufficient condition for a torsion group to be a direct sum of cyclic groups. For arbitrary groups, however, no satisfactory condition is known so far. The known criteria refer to a selected set of elements, namely, the basis of the group, and so these can be much better described as conditions as to whether or not a given or selected subset is a basis. For criteria on the existence of a basis we may refer to Fuchs [ I ] or [7]. However, in the case of countable torsion-free groups there is a useful criterion.

Theorem 19.1 (Pontryagin [ I ] ) . A countable torsion-free group is free if and only if every subgroup of Jinite rank is free. Equivalently, for every integer n, the subgroups of rank 5n satisfy the maximum condition. Because of (14.9, the necessity is evident. In order to establish sufficiency, let A = ( a l , . . . , a,, , . . .) be a countable torsion-free group all of whose subgroups of finite rank are free. Define A , as the set of all a E A which depend on { a , , . . . , a,,}, with 0 adjoined. Then A , is a subgroup of rank s n . Clearly, I ( A , + ~5 ) r(A,) + 1; therefore either A is of finite rank [in which case the assertion is trivial] or there is a subsequence B, of the A,, such that r(B,) = n and A is the union of the ascending chain 0 = B, < B , < B , < . . . < B,, < . . . . Now B,+,/B,, is torsion-free [because A / A , is], of rank 1, and finitely generated; hence B,+,/B, E Z. From (14.4) we get

1x1

B,+, = a,, < b n + 1 ) for some b,+l. This show> that b, . .., b,, ... generate the direct sum @,(b,,), whence A = Q,(b,). In view of (15.5), the second formulation is equivalent to the first 0ne.m

It should be noted that the preceding theorem fails to hold for groups of larger cardinalities, as will be shown by the next theorem which is of independent interest.

94

111.

DIRECT SUMS OF CYCLIC GROUPS

For a cardinal K,, call a group KC-free,if all of its subgroups of cardinality less than K, are free. Theorem 19.2 (Baer [2], Specker [ 11). A direct product of an infinite set of infinite cyclic groups is K,-free, but not free. Write A = E I ( a i ) where I is an infinite set and ( a i ) r Z for each i. First we prove that every finite set {b,, . .., bm}c A can be embedded in a finitely generated direct summand of A . If m = 1 and b, # 0, write b, = (. . . , ni ai , . . .) with ni E Z. If there is a j E I with In,] = 1, then we have A = (6,) @ A , where A , consists of all elements of A with j t h coordinates 0. If the minimum n of lnil with ni # 0 is greater than 1, then set n, = q i n + ri (0 5 ri < n) and define c1 = (..., q i a i , ...), c2 = (..., r i a i , ...) in A so that b, = nc, + c 2 . There is some j with lqjl = 1 and r, = 0, thus A = (cl) @ A , where c2 E A , with coefficients lril < n. By induction, A , has a finitely generated direct summand B' containing c 2 , and so (cl> @ B' is a direct summand of A containing b,. Assume m > 1 and A has a finitely generated direct summand B such that b,, . . . , b m - , E B. We may, in addition, suppose that A = B a C where C is the direct product of almost all ( a i ) . Setting 6, = b + c ( b E B , c E C ) and embedding c in a finitely generated direct summand C' of C , B @ C' will be a finitely generated direct summand of A that contains all of b,, ..., 6,. If G is now a countable subgroup of A , then a maximal independent set of a subgroup G' ( 5G) of finite rank is contained in a finitely generated direct summand B of A , and by torsion-freeness, G' 5 B. Now (19.1) proves G free; thus A is K,-free. If A were free, then all the subgroups of A would be free, so it suffices to exhibit a nonfree subgroup H of A . Let H be the subgroup of A' = f l ~ , ( a i ) consisting of all b = (n,a,, . . . , n, a i , . . .) such and for any positive integer m and a fixed prime p , almost all coefficients ni are divisible by pm. This H is clearly of the power of the continuum that contains @(ai>. Since each coset of H mod p H contains some element of @ ( a , ) , H/pH is countable. The comparison of the cardinalities of H and H/pH shows that H cannot be free.0

ni

An immediate consequence is : Corollary 19.3 (K. Stein). Any countable group A can be written in the form A=N@F

where F isfree and N has no free quotient groups. N is uniquely determined by A . Define N as the intersection of the kernels of all homomorphisms q : A + 2. Then A / N is isomorphic to a subgroup of a direct product of infinite cyclic groups, and so it is free by (19.2). N is a direct summand owing to (14.4).

NOTES

95

The uniqueness of N follows from the observation that if M is any direct summand of A without free quotient groups, then its projection into F must vanish by (14.5), so M 5 N.0

EXERCISES 1. A countable group is a direct sum of cyclic groups if and only if every finite set of elements can be embedded in a finitely generated direct summand. be a generating system of a group A where 2. (Szele [3]) Let L = each a, is of infinite or prime,power order. L is a basis of A if every finite subsystem a,, ..., ak of L satisfies: ( a , , ..., ak) = (bl, ..., b k ) with o(bi)= co or p' implies min(o(al), . . . , o(aa))5 min(o(b,), . . ., o(bk)). 3. A subset of a torsion-free group A is a basis of A if and only if it is a minimal generating system such that, for a finite subset {al, . . . , ak}, U E Adepends on {al, . . . , ak} implies G E (ul, -..,ak). 4. The same with " minimal generating system " replaced by " maximal independent system." 5. (a) The intersection of direct summands of a countable free group is again a direct summand. [ H i n t : cf. Ex. 4 in 14; apply (19.2) t o the quotient group mod intersection.] (b) The same fails to hold for a free group of the power of the continuum. 6. In any presentation of Z K othere are uncountably many defining relations. 7. Given any cardinal in 2 2O ' , there exists a n '&,-free group of cardinality m which is not free. NOTES The existence of free objects in the category of all abelian groups is fundamental. Though in homological algebra, projectivity is predominant, in abelian group theory freedom seems to be prevailing. Fortunately, for abelian groups, freedom and projectivity are equivalent, while for modules, the projectives are exactly the direct summands of free modules. They are free over local rings (see Kaplansky [3]) or over polynomial rings with a finite number of noncommuting indeterminates with coefficients in a commutative field (see P. M. Cohn [ J . Algebra 1 (1964), 47-69]). Theorem (14.5) holds for modules over left principal ideal domains. Submodules of projectives are again projective if and only if the ring is left hereditary, i.e., all left ideals are projective. (14.1) holds over commutative rings or under the hypothesis that at least one of m and n is infinite. There exist, however, rings R such that all free R-modules # O with finite sets of generators are isomorphic. It is perhaps worth while pointing out that every R-module is free if and only if R is a field, and every R-module is projective exactly if R is a semisimple Artin ring.

96

111.

DIRECT SUMS OF CYCLIC GROUPS

A module M is quasi-projective if, for every epimorphism p: M + N and every homomorphism $: M + N, there exists an endomorphism # of M such that &h = Quasiprojective modules were studied by L. E. T. Wu and J. P. Jans [I[[. J . Math. 11 (1967), 439-4481. An abelian group is quasi-projective exactly if it is either free or a torsion group every p-component of which is a direct sum of cyclic groups of the same order p"; see L.Fuchs and K. M. Rangaswamy [to appear in Bull. SOC.Math. France]. There are numerous generalizations of the results in 15. I . Kaplansky [ J . Indian Math. SOC. 24 (1960), 279-2811 proved that, for commutative domains R, the torsion parts of all finitely generated R-modules are direct summands exactly if R is a Priifer domain. E. M a t h [Trans. Amer. Math. SOC. 125 (1966), 147-1791 made agood approach toward thedescription of commutative domains over which finitely generated torsion modules split into direct sums of cyclic modules. Supposing R is a commutative Noetherian ring, A. I. Uzkov [Mat. Sb. 62 (1963), 469-4751 showed that every finitely generated R-module is a direct sum of cyclic modules if, and only if, R is a principal ideal ring. Unless the ring is left Noetherian, finitely generated modules are different from finitely presented modules (where the finiteness of a set of defining relations is also assumed); these are somewhat better manageable, and it is still true that every module is the direct limit of finitely presented ones. For finitely presented R-modules, a direct sum decomposition into cyclic modules holds if, and only if, R is an elementary divisor ring, i.e., every matrix over R can be brought to a diagonal form by left and right multiplications by unimodular matrices (I. Kaplansky [Trans. Amer. Math. SOC.66 (1949), 4-91]). In this case (15.4) holds true. Little attention has been paid so far to infinite direct sums of cyclic modules; even the uniqueness of decompositions has not been dealt with. One of the difficulties lies in the fact that maximal independent systems no longer yield invariants for the modules, except for a restricted class of rings; cf. KertCsz [Acfa Sci. Math. Szeged. 21 (1960), 260-2691 and L. Fuchs [Annales Univ. Sci. Budapest 6 (1963), 71-78].

+.

Problem 9. Characterize the groups in which the intersection of two direct summands is again a summand. Cf. 14, Ex. 4 and 19, Ex. 5

Problem 10. For which ordinals 0 d o there exist &-free groups which are not KO+,-free? Problem 11. Investigate K-direct sums of cyclic groups.

IV DIVISIBLE GROUPS

This chapter is devoted to one of the most important classes of abelian groups: the divisible groups. Divisible groups are extremely easy to recognize, and they have a number of prominent properties which are characteristic for them. One of their outstanding features is that they coincide with the injective groups which embody a concept dual to projectivity, and thus they are direct summands in any group containing them. The structure of divisible groups is completely known, so that the theory of divisible groups is as satisfactory as one could expect at the present status of algebra. T h e concluding topic for this chapter is concerned with a remarkable duality between maximum and minimum conditions.

20. DIVISIBILITY

Since multiplication o f group elements by integers makes sense, it is natural to introduce divisihility of group elements by integers. We shall say that the element a o f the group A is divisible by the positive integer n, in symbols: n I a, if the equation (1)

nx=a

(UEA)

is solvable in A ; that is to say, A contains an element b such that x = b is a solution of (1). It is clear that the solvability of (1) is equivalent to a E nA. Let us list some elementary consequences of our definition.

+

(a) If x = b is a solution of (l), then the coset b A[n] is the set of all solutions of (1). (b) If A is torsion-free, then (1) has at most one solution.

97

98

Iv.

DIVISIBLE GROUPS

(c) If (n, o(a)) = 1, then (1) is always solvable. For if r, s are integers such that nr + o(a)s = 1, then x = ra satisfies nx = nra = nra + o(a)sa = a. (d) m I a and n I a imply [m, n] I a. Let r , s be integers such that mr + ns = d = (m, n), and let b, c E A satisfy mb = a, nc = a. Then [m, n](rc + sb) = mn d-'(rc + sb) = md-lra nd-lsa = a. (e) nl a and n I b imply n I a f b. (f) If A is adirect sum, A = B e C, thennla = b + c ( b E B , C EC)if,and only if, n I b and n I c. The same holds for infinite direct sums and direct products. (g) For every homomorphism ct : A + B, n I a in A implies n I cta in B. (h) If p is a prime, pk I a if, and only if, k 5 h,(a).

+

A group D is called divisible if n I a for all a E D and all positive integers n. Thus D is divisible if and only if nD = D for every positive n. The groups Q, Z(p"), Q / Z are examples for divisible groups, but a direct sum of cyclic groups is not divisible [unless 01. A group D is said to bep-divisible ifpkD= D for every positive integer k. Since p k D = p . p D , it is obvious that p-divisibility is implied by p D = D. The last equality is equivalent to the fact that every element of D is divisible by P. (A) A group is divisible i f and only i f it is p-divisible for every prime p. If p D = D for every prime p and n = p l * . p,., then nD = p l " . p r D = D. Thus divisibility is equivalent to the fact that every element is of infinite p-height at every prime p. ( B ) A p-group is divisible ifand only if it isp-divisible. In view of (c), for a p-group D we always have qD = D, whenever the primes p , q are different. ( C ) I f in a p-group D, every element of order p is of infinite height, then D is divisible. Let a E D be of order p". We prove by induction on n that p I a. For n = 1 , this being a consequence of the hypothesis, assume n > 1, and that we have proved this for elements of order 0. Then A is the direct sum of cocyclic groups. [Hint: p"A is divisible.] 10. Show that any two direct decompositions of a divisible group have isomorphic refinements. 11. A group A is called quasi-injective if every homomorphism of every subgroup of A into A can be extended to an endomorphism of A . Prove that a quasi-injective group is either injective or is a torsion group whose p-components are direct sums of isomorphic cocyclic groups. 24.

THE DIVISIBLE HULL

Free groups are universal in a certain sense, namely, every group is an epimorphic image of some free group. The next result shows that divisible groups are universal in the dual sense:

Theorem 24.1. Every group can be embedded as a subgroup in a divisible group. The infinite cyclic group Z can clearly be embedded in a divisible group, namely, in Q. Hence every free group F is embeddable in a direct sum of copies of Q, i.e., in a divisible group. Given an arbitrary group A , we may write A F/N for a suitable free group F. If we embed F i n a divisible group D , then A will be isomorphic to the subgroup FIN of the divisible group DIN.0 This theorem can be improved by establishing the existence of a minimal divisible group containing a given group. To prove this, we begin with the following lemmas.

Lemma 24.2. A subgroup B of A is essential i f and only i f a homomorphism + G with an arbitrary group G is necessarily monic whenever ci I B : B + G is a monomorphism. If B is an essential subgroup of A , and ci1 B is a monomorphism, then Ker ci n B = Ker(ci I B ) = 0 implies Ker ci = 0. Conversely, if B is not essential, and if C , 0 < C 5 A , satisfies C n B = 0, then the canonical epimorphism ci : A + A / C is not monic, though ci I B is a monomorphism.0 ci

:A

Given A , call the divisible group E containing A minimal divisible if no proper divisible subgroup of E contains A .

24.

107

THE DIVISIBLE HULL

Lemma 24.3. A divisible group E containing A is minimal divisible exactly if A is an essential subgroup of E. If E is not minimal and D is a proper divisible subgroup of E containing A , then we may write E = D @ C with C # 0. Since A n C 5 D n C = 0, A is not essential in E. Conversely, if A is not essential in E, then there is a cyclic group C = (c) # 0 in E such that A n C = 0. We may assume, without loss of generality, that o(c) = co or pk. Then C can be embedded in a subgroup B of E such that B z Q or Z(p"), and by (21.2), E = D 0B for some group D containing A. Thus E is not minimal.0 Theorem 24.4 (Kulikov [2]). Erery divisible group containing A contains a minimal divisible group containing A . Any two minimaldivisiblegroups containing A are isomorphic over A . Let D be divisible and contain A. The divisible subgroups of D that are disjoint from A form an inductive set, hence there exists a maximal member M in this set. By (21.2), we have D = M @ E with A 5 E. Clearly, E is divisible, and the maximality of M guarantees that E cannot have any proper direct summand and, hence, any proper divisible subgroup still containing A. Therefore, D contains a minimal divisible subgroup E containing A. If E l , E, are two minimal divisible groups containing A , then because of (21. l), the identity automorphism I A of A can be extended to a homomorphism r] : E , ---t E, . Since qEl is divisible and contains A , q is an epimorphism. By (24.3), A is an essential subgroup of El, and since q I A = l,, (24.2) shows r] monic. Consequently, q is an isomorphism of E, with E , leaving A elementwise fixed.n By the last theorem, we are justified in calling a minimal divisible group E containing A the divisible hull [injective hull] of A. Clearly, (24.3) shows that

r,(E)

= r,(A)

and

r,(E)

= r,(A)

for every prime p.

Thus the structure of the dirisible hull of a group A is completely determined by the ranks of A . The results of this section enable us to establish the converses of (21.1) and (21.2). Theorem 24.5. For a group D , the follo\+3ingconditions are equivalent : (i) D is dizisible; (ii) D is injective; (iii) D is a direct summand of ewry group containing D

We need only verify that (iii) implies (i). Embed D in a divisible group E. Then (iii) implies that D is a direct summand of E, hence (i) holds.0

Iv.

108

DIVISIBLE GROUPS

It is convenient at times to call an exact sequence O+A+D+D’+O with D [and hence D’] divisible, whose existence for every A is guaranteed by (24.1), a dirisible or injectire resolution of A . We finally introduce a theorem, mainly because it can now be proved, and its proof is based on divisible groups; it will be convenient to have it at disposal for easy reference. Proposition 24.6. If‘C is a subgroup of a group B such that BIC is isomorphic to a subgroup H of G, then there existsagroup A containing Bsuch that AIC z G. Let D be a divisible group containing B. Now D j C contains BIC, and if BjC cannot be extended in D j C to a group isomorphic to G, then we can always find a divisible group E such that in D / C @ E this embedding is possible. If A j C is isomorphic to G and contains B/C E H , then A ( 5D @ E ) is a group with the desired properties.0

The situation of (24.6) is shown by the commutative diagram O+C+B+H-+O

where the rows are exact, and the vertical maps are injections.

EXERCISES For a torsion-free group A , let E be the set of all pairs (a, m) with a E A , a n integer >0, such that

in

(a, in) = (h, n )

and (u, n7)

if and only if mb

= nu,

+ ( h , n ) = (nu + nib, nm).

Show that E is a minimal divisible group containing the image of the monomorphism a ++(a, 1) (a E A ) . The divisible hull of A is the divisible hull of a subgroup B of A if and oiily if B is essential in A . (a) A divisible group D containing A is minimal exactly if DIA is torsion and A contains the socle of D. (b) I f D is a divisible hull of a p-divisible group A , then D / A has trivial p-component. (KertCsz [2]) (a) I f A is an endomorphic image of every group containing it, then A is divisible. (b) If A is a direct summand of every group containing it as an endo-

25.

5.

6.

7. 8*.

9.

109

FINITELY COGENERATED GROUPS

morphic image, then A is divisible. (KertCsz [2]) Let A be such that if A is an endomorphic image of a group B, then B contains a direct summand isomorphic to A . Show that A is a direct sum of a divisible and a free group. If A is a direct summand in every group G which contains A such that G / A is quasicyclic, then A is divisible. (Dlab [2]) Every group is the Frattini subgroup of some group. (Charles [2], Khabbaz [2]) A subgroup A of a divisible group D is the intersection of divisible subgroups of D if and only if for every prime p , A [ p ] = D [ p ] implies p A = A . Let A , (n = 0, I t l , +2, . . . ) be a countable system of groups. Verify the existence of divisible groups D, and a sequence *

’ . -P D,-

-,.

1a n - L + . ~ n ~ + ~ n + l

such that CX,,~,,-~ = 0, and Ker cc,/Im A , for every n. 10. (Szele [2]) Call a E A algebraic over a subgroup B of A if a = 0 or ( a ) n B # 0. If every a E A is algebraic over B, we call A algebraic over B. (a) A is algebraic over B if and only if B is an essential subgroup of A . (b) A is a maximal algebraic extension of B [i.e., no group properly containing A is algebraic over B ] exactly if A is a divisible hull of B. 11. (Szele 121) (a) If B 5 A , then there exists an algebraic extension C of B in A which is maximal among the algebraic extensions of B in A . (b) Conclude, by making use of (a) and Ex. 10, that every divisible group containing B contains a divisible hull of B. 12. (E. A. Walker [4]) G is said to be an n-extension of A , if G is an essential extension of A such that A = nG. (a) If G is an n-extension of A , then there is a divisible hull D of A containing G. (b) If both G and G’ are n-extensions of A and G 5 G’, then G = G’. (c) Prove the existence of an n-extension of any given A by proving that ( D / A ) [ n ]= G / A defines a group G with nG = A . (d) Any two n-extensions of A are isomorphic over A . (e) Every group H with nH = A contains an n-extension of A . 25. FINITELY COGENERATED GROUPS

The duality we have noticed between free and divisible groups suggests a concept dual to finitely generated groups. A system L of elements of a group A is called a system of cogenerators if, for every group B, every homomorphism d, : A --* B such that L n Ker I$ = fa or 0 must be a monomorphism. This is evidently equivalent to the condition

110

Iv.

DIVISIBLE GROUPS

that every nonzero subgroup of A contains a nonzero element of L. Clearly, the subgroup ( L ) must be an essential subgroup of A , and an essential subgroup is always a system of cogenerators. What has been said at the introduction of cocyclic groups [in 31 indicates that cogenerators are dual to generators. A group will be called finitely cogenerated if it has a finite system of cogenerators. The following result is an analog of (15.5) and points out a beautiful duality between maximum and minimum conditions. Theorem25.1 (Kurosh [l], Yahya [I]). For a group A , the following conditions are equivalent : (i) (ii) (iii) (iv)

A isfinitely cogenerated; A is an essential extension of afinite group; A is a direct sum of a finite number. of cocyclic groups; the subgroups of A satisfy the minimum condition.

Assume (i) and let L be a finite system of cogenerators for A . No element of A is of infinite order, for otherwise we could select in the cyclic group generated by an element of infinite order a subgroup # O excluding all the nonzero elements of L. Thus L is a finite system of elements of finite order, whence ( L ) is finite. A being an essential extension of ( L ) , (ii) follows. Next assume (ii), i.e., A is an essential extension of a finite subgroup B . Clearly, A is torsion with a finite number of p-components, and in order to prove (iii), we may suppose A , B are p-groups. Considering that A [ p ] = B [ p ] is finite, for a fixed a E A , the equation px = a may have but a finite number of solutions in A. If h(a) = co, then the heights of the solutions xl, ... , x k cannot be all finite, for ify E A satisfiesp"y = a, thenp"-'y is among xl, * . . , xk. Hence, we conclude that starting with a E A [ p ] of infinite height, we can ascend to obtain a quasicyclic subgroup of A . The union D of the quasicyclic subgroups of A is divisible, hence A = D @ C. Since C [ p ] is finite, there is a finite maximal height m of its elements, and so p m + ' C= 0. Thus A is a direct sum of cocyclic groups where the number of summands must be finite owing to the finiteness of the socle. We next show that (iii) implies (iv). If r ( A ) = 1, i.e., A = 2bk)with 1 k 5 03, then the statement is evident. For r(A) = n > 1 we use induction, and assume the minimum condition in groups of rank Sn - 1. Let A = 2 ( p k )0 B where r(B) = n - 1, and let GI 2 G, 2 . . . be a descending chain in A . Then B n G, 2 B n G2 1 . . ., and from some index m on, we have B n G, = B n G,+, = . . * . From

G,/(B n C,) = C J ( B n Gi) ( B + G i ) / B5 A / B = Z(pk) (i 2 m) we infer that the G,/(B n C,), and hence the G i , are equal from some index on. This proves (iv).

25.

111

FINITELY COGENERATED GROUPS

Finally, assume (iv). A contains no element a of infinite order, for (2"a) (n = 0, I , 2, . . -) cannot be a properly descending infinite chain in A. Since the socle of A cannot be an infinite direct sum, S ( A ) is finite, and hence A has a finite system of cogenerators.0 We observe that from the equivalence of (i) and (iv) it results that quotient groups of finitely cogenerated groups are finiteIy cogenerated. Also, notice that (ii) is equivalent to the finiteness of the socle. To conclude, we prove the following simple result. Proposition 25.2. Let a,, . . . , a, be a jinite set of nonzero elements in A , and M a subgroup of A that is maximul ,r*ithrespect to the property of excluding each of a,, . . . , a,. Then AIM satisfies the minimum condition for subgroups. If n = 1, then AIM is cocyclic. Every subgroup of A which contains M properly must contain at least one of a,, . . . , a,. In other words, every nonzero subgroup of AIM contains one of a, M , ..., a, + M , that is a, M , ... , a, + M is a system of cogenerators of AIM. The preceding theorem completes the proof.0

+

+

Notice that (25.2) implies that every group is Hausdorff in the Prufer topology. EXERCISES 1. A subdirect sum of a finite number of groups with minimum condition satisfies the minimum condition. 2. If a subgroup B of A and the quotient group AIB both satisfy the minimum condition, then so does A. 3. A p-group satisfies the minimum condition if and only if it is of finite rank. 4. If u] is an endomorphism of a group with minimum condition such that Ker u] = 0, then u] is an automorphism. 5. Find the structure of groups in which there are but a finite number of elements of any fixed order. 6 . (a) Characterize the groups in which the set of finitely generated subgroups satisfies the minimum condition. (b) Dually, describe the groups in which the set of subgroups with minimum condition satisfies the maximum condition. 7. The set of endomorphic images satisfies the minimum condition if and only if the group is a direct sum of a finite number of groups Q and

Z(pk)(k = 1,2, ... , a).[Hint: nA.] 8. The set of fully invariant subgroups satisfies the minimum condition if and only if the group is a direct sum of groups Q,Z(p") for finitely many p and Z(pk)with pk 1 rn for a fixed m.

Iv.

112

DIVISIBLE GROUPS

NOTES Injectivity together with the existence of injective hulls was discovered by Baer (31. H e also proved that for the injectivity of an R-module M , it is necessary and su.%cient that every homomorphism from every ieft ideal L of R into M extends to an R-homomorphism R + M . This extensibility property, with L restrictzd to principal left ideals of R, is perhaps the most convenient way to define the divisibility of an R-module M . It is then immediately clear that an injective module is r.ecessarily divisible. For modules over ;ommutative domains R, the equivalence of injectivity and divisibility characterizes the Dedekind domains (see Cartan and Eilenberg [I]). For torsion-free modules over Ore domains divisibility always implies injectivity. Epimorphic images of injective R-modulss are again injective if and only if R is left hereditary. It is worthwhile noting that a commutarivs domain is hereditary exactly if it is a Dedekind domain. The results in 22 prevail for injective modules in general. Let us note that the semisimple Artin rings [the regular rings in the sense of von Neumann] are characterized by the property that all modules over them are injective [divisibie]. It is an easy exercise to show that over left Noetkiizr, rings every module has a maximal injective submodule. This is not necessarily a uniq;iely determined submodule, unless the ring is, in addition, left hereditary. E. M a t h [fac.J. Mutiz. 8 (1958), 51 1-5281 and Z. Papp [Publicutiones Math. Debrecerz 6 (1959), 31 1-3271 proved that every injective module over a left Noetherian ring R is a direct sum of directly indecomposable ones. If, in addition, R is commutative, then the indecomposable R-modules are in a one-to-one correspondence with the prime ideals P of R, namely, they are the injective hulls of R/P (for R = Z take P = ( O ) or ( p ) ) ; cf. M a t h [loc. cit.]. I t is a remarkable fact that direct sums (and direct limits) of injectives are again injective if and only if the ring is left Noetherian-as was pointed out by Matlis and Papp.

Problem 12. Which groups are (a) projective, (b) injective over their endomorphism rings? (c) Find the projective, injective. and the weak dimensions of a group over its endomorphism ring. See A. J. Douglas and H. K. Farahat [Monatshefre Math. 69 (1965), 294-3051 and F. Richman and E. A. Walker [Math. Z . 89 (1965), 77-81].

V PURE SlJBGROUPS

The notion of pure subgrGups is due to Prufer [21; it has recently become one of the most useful concepts in abelian group theory. The notion of pure sujgroups is intermediate between subgroups and direct summands, and i t reflects a way ir. ..,hich a subgroup is embedded in the whole group. This is sufficiently general to gusrantee that there is an adequate supply of pure subgroups, and at the same time piire subgroup!: share a number of properties which are easy to handle. Their importance lies also in the methodological role they play in proving the existence of direct summands; namely, the existence of pure subgroups of one kind or another is easily established, and several criteria assure the direct summand character of certain pure subgroups. If the exact sequences a r e restricted to those which we are going to call pure-exact, then the new, remarkable notion of pure-injectivity arises [which will be the topic of Chapter VII].

26. PURITY

A subgroup G of A is called pure, if the equation nx = g E G is solvable i n G, whenever it is solvable in the whole group A . This amounts t o saying that G is pure in A if 17 I g in A implies n I g i n G. As n I g in G is equivalent to the inclusion g E nG, we see that C is pure in A if and only if

(1)

nG

=

G n nA

for every n E Z.

If both A and G are equipped with their Z-adic topologies, then ( I ) evidently imglies that for u pure subgroi4p, the relutim 2-adic topology arid its own Z-adic topology are the same. A natural generalization of purity is p-purity. A subgroup G of A is p-pure ( p a prime) if (2)

pkG = G np k A

for k

=

I , 2,

..*,

113

114

v.

PURE SUBGROUPS

or, in other words, the p-heights of elements of G are the same in G as in A . For p-pure subgroups, the relative p-adic topologies coincide with their own p-adic topologies. If G is p-pure in A for every prime p , then C is pure in A . In fact, if n = pi' * py is the canonical representation, then

nG

= p;'G n . * - n p r G = (G n p*,'A)n ... n (G n @A) = G n (p;lA n ... n prkA)= G n nA.

We next list some useful facts concerning pure subgroups. (a) Every direct summand is a pure subgroup. 0 and A are (the trivial) pure subgroups of A . (b) The torsion part of a mixed group and its p-components are pure subgroups [these fail to be, in general, direct summands]. (c) The subgroups of Q and Z(p") have no pure subgroups except for the trivial ones. (d) If A / G is torsion-free, then C is pure in A . In fact, nu = g E G (n # 0) for a E A implies a E G. (e) In torsion-free groups, intersections of pure subgroups are again pure. Indeed, an equation nx = g has at most one solution; therefore, if it is solvable in A , then its unique solution belongs to every pure subgroup containing g. In view of this, in a torsion-free group A , to every subset S there exists a minimal pure subgroup containing S, namely, the intersection ( S ) , of all pure subgroups of A that contain S ; it may be called the pure subgroup generated by S. It is easy to check that ( S ) , is the set of all elements of A which depend on S. (f) Purity is an inductive property. be a chain of pure subgroups G i of A , and let G Let G, 5 * * G i5 be their union. If nx = g E G is solvable in A , and i f j is an index with g E G j , then nx = g is solvable in G j , and hence a fortiori in G. Similarly, p-purity is inductive. (g) A p-pure p-subgroup of a group is pure. This follows at once from the equation qG = G which holds for every p-group G and every prime q # p. (h) If A is p-group and if the elements of order p of a subgroup G have the same height in G as in A , then G is pure in A . We use induction on e(g) = n to prove that every g E G has the same height in G as in A . For e(g) = 1 , this being our assumption, suppose this holds for elements of exponent 0, proving the purity of C in A. The statement (ii) follows from the equalities

n(B/C)= nB + C = ( B n H A )+ C = B n (nA

+ C ) = B n n(A/C).

If we assume the hypotheses of (iii), then let nu = b E B for some a E A and integer n > 0. Now n(a C) = b C and the hypothesis imply that for some b' E B, n(b' + C ) = b + C. From nb' = b + c (c E C) we get n(b' - a) = c; thus nc' = c for some c' E C. Finally, n(b' - c') = b with b' - c' E B completes the proof.0

+

+

Owing to (ii) and (iii), the natural correspondence between subgroups of A / C and subgroups of A containing the pure subgroup C preserves purity. A result of another nature is: Proposition 26.2 (T. Szele). Every infinite subgroup can be embedded in a pure subgroup of the same power and every finite subgroup in a countable pure subgroup. Given a subgroup B of A , IBI = in, consider all equations nx = b E B which are solvable in A . For each such equation we adjoin a solution an,bE A to B in order to get a subgroup B, of A in which all such equations over B are solvable [i.e., B, is generated by B and all these an,b]. Then we repeat this process with B replaced by B, to obtain B, in which all equations with right members in B, solvable in A are solvable. The union G of the Bi ( i = 1, 2, . ..) is a pure subgroup of A , since nx = g E G is solvable in B,+, if g E Biand if

v.

116

PURE SUBGROUPS

the equation has a solution i n A . Clearly, [GI 5 inKO whence both parts of the assertion follow.0 For torsion-free groups, (26.2) can be improved: every subgroup can be embedded in a pure subgroup of the same rank, namely, in the pure subgroup generated by it. Purity can be defined for p-adic modules by replacing natural integers n in (1) by p-adic integers. As multiplication by units does not change submodules in the whole, only the powers ofp must be taken into consideration and therefore, for p-adic modules the definition of purity [and p-purity] is given by (2).

EXERCISES

+

1. If C n H and C H are pure subgroups of A, then so are G and H . 2. (a) Neither the intersection nor the union of direct summands need be pure. (b) If G is a pure subgroup of each member of a chain . . . 5 B , . . . , then G is pure in U S , . 3. The K-direct sum of groups is pure i n the direct product. 4. If G is pure in A , then ,7G is pure in /?A. 5. If T is the torsion part of A , and if G is pure in A , then T + G is pure in A . 6. If G is a pure subgroup of the group A , then: (a) G' = G n A' (first Ulm subgroups); (b) (G + A ' ) / A ' is pure in @ A ' ; (c) G 5 A' implies G is divisible. 7. Give an example for a nonpure subgroup i n a group, such that its own 2-adic topology is the same as the relative Z-adic topology. 8. The only pure, essential subgroup of A is A itself. 9. A group is pure in every group containing it exactly if i t is divisible. 10. If in a countablep-group A , the elements of infinite height form a pure subgroup, then A is a direct sum of cocyclic groups. [Hint: Ex. 6 (c), (17.3), and (23.1).] 11. For every pure subgroup B of A , there exists a well-ordered ascending chain of pure subgroups of A :

B

= B,

< B , < * . . < B, < ... < B, = A

(a < T )

u,, p m ; we use them in order to establish a correspondence f : ( a i ) H ( a j ) in the set D of direct summands in a fixed direct decomposition of B: .f is subject to the following conditions: (i) j i s one-to-one between D and a subset 3‘ of D; (ii) if$ ( a i ) t+ ( a j ) , where e(ai) = k i , e(aj) = k j , then k j 2 2ki. The choice of m guarantees that for every k m , D contains fin r(B) many summands ( a , ) of order &pk, so such an f does exist. Now define a mapping E on a subset of B as follows: (a) E U = U (b) &aj= ai (c) &aj= 0

ifaEB,@...@B,; if ( a j ) E D’and f:(a,) H ( a j ) ; if ( a j ) $ D’.

Evidently, E induces a well-defined endomorphism of B onto itself which may again be denoted by E. This extends to an endomorphism q : A + B as follows.

37.

THE ULM SEQUENCE

153

If a E A is of order pr, then write A = B , @ * - . @ B, @ A , (n 2 2r), and a = b + c with b E B, @ . * * @ B , , c E A , , and define r]a = Eb. This definition

does not depend on the choice of n ( 2 2r), for the B,-component of a in the decomposition B, @ . . . @B, @ A , is divisible by pS-', and therefore it is carried into 0 by E whenever s 2 2r. That q preserves sums is evident, hence r] is an endomorphism of A onto B . 0

Corollary 36.2. r f C is a pure subgroup of a p-group A , and if it is a direct sum of cyclic groups, then C is a n endomorphic image of A . A basis of C is p-independent in A , hence it extends to a p-basis of A. By (36.1), the subgroup B generated by this p-basis is an endomorphic image of A , and so is C as a direct summand of B . 0

EXERCISES (Szele [7]) Every p-group of cardinality Stn (an infinite cardinal) is an epimorphic image of a p-group A if and only if m S fin r(B) where B is a basic subgroup of A . Let A be torsion-free. Its p-basic subgroup B is an endomorphic image of A if and only if A has a direct summand which is a free group of rank Under which conditions is a basic submodule of a p-adic module A an endomorphic image of A ? [Hint: cf. Ex. 2.1 State a necessary and sufficient condition for the existence of an epimorphism B + A, where B is basic in the p-group A . 37. THE ULM SEQUENCE

We have already introduced the Ulrn subgroups and the Ulm factors of a group [see 61, but we have failed to state some important results on them which have numerous consequences in our further developments and which at the same time shed more light on the structure of groups in general. Since we are now in a position to establish the most important properties of the Ulm sequence, we turn our attention to this important invariant of a group. Let us recall that, for an ordinal 0,the 0th Ulm subgroup A" of a group A was defined inductively as follows: A' = A, A" " = n , n A " and, for a limit ordinal 0,A" = A P .The Ulm length u(A) of A is the least ordinal z such that A r t l = A'. Clearly, this T exists and does not exceed IAl. It is also evident that A' is the maximal divisible subgroup of A . Thus, if we restrict ourselves to reduced groups A-which we may do without loss of generalitythen A' = 0, and we arrive at a well-ordered descending chain of subgroups of A : A = A' > A' > A 2 > .. . > A " > A U t 1 > .. . > A' = 0.

nP 0 such that nA is contained in the direct sum of aJinite number of the C i . If the conclusion fails, then there exist a properly increasing sequence of natural numbers n,, n j , ... with n j I n j + l and groups B j , each being a direct sum of a finite number of C i, such that the Bj generate their direct sum in @Ci and i

j- 1

n j A n @ Bk < n j A k= 1

fl

@ Bk k=l

( j = 1, 2, "').

166

VII.

ALGEBRAICALLY COMPACT GROUPS

Let aj be in the right, but not in the left member; then a j - l has 0 and aj has nonzero coordinate in Bj . Therefore, the Cauchy sequence a,, * , aj , ...E A cannot have a limit in @ C i

-

.o

Corollary 39.10. If A = @ E I Ci is a direct decomposition of a complete group A, then all the C, are complete groups, and there is an integer n > 0 such that nCi = 0 for ulmost all i E I. The first statement results from (1 3.4), while the second is an immediate consequence of (39.9).0

EXERCISES 1. Let m > 0 be an integer. Show that A is complete if and only if mA is complete. 2. (Kaplansky [2]) If C is a pure subgroup of a complete group A , then the closure of C in A is a direct summand of A. 3. Describe the Z-adic completions of the following groups: Z,

Q , , Q'"),

6Z(P">, 0%).

n= 1

4. The 2-adic completion of a direct product is the direct product of the 2-adic completions of the components. 5. (a) Show that the p-adic completion of a group A is isomorphic t o b,A/pkA. (b) Prove that the p-adic completion is always a p-adic module. (c) If A is complete in its p-adic topology, then qA = A for every prime 42P. 6. Prove (39.7) and (39.8) for p-adic completions [replace pure-exactness by p-pure-exactness]. 7. Let B = @, Bn where Bn = @Z(p"). Show that if B* = B,, and CIB is the maximal divisible subgroup of B*/B, then C i s a p-adic completion of B. 8. If A is complete in its p-adic topology and B is a p-basic subgroup of A, then every homomorphism B -+ A extends uniquely to an endomorphism of A . 9. If B is a basic submodule of the p-adic module A, then every finitely generated direct summand of B is a direct summand of A, too. 10. Prove that neither J = J , nor n , Z ( p ) is a direct sum of infinitely many subgroups #O. 11. Let A , ( ~ E Z )be algebraically compact groups. Find a necessary and sufficient condition for @ A i to be algebraically compact.

nn

n,,

40.

167

THE STRUCTURE OF ALGEBRAICALLY COMPACT GROUPS

12. The inverse limit of nonreduced algebraically compact groups need not be algebraically compact. [Hint: use the idea of 21, Ex. 6 to get an unbounded p-group for inverse limit.] 13. (Fuchs [13]) The inverse limit of groups satisfying the minimum condition is algebraically compact. [Hin?:13, Exs. 6 , 7 and 38, Ex. 10.1 40. THE STRUCTURE OF ALGEBRAlCALLY COMPACT GROUPS

We have seen a number of useful properties of algebraically compact groups, and now we have come to the structure theorem on algebraically compact groups. Since the structure of divisible groups can be described in terms of cardinal invariants, there is no loss of generality in restricting our discussion to the reduced case. A further reduction is possible in view of: Proposition 40.1 (Kaplansky [2]). For a reducedgroup A to be algebraically compact, it is necessary and sufficient that it is of the f o r m

,

[the product being extended oZjer all distinct primes p] tihere each A is couiplete in its p-adic topology. The A , are uniquely determined by A . First, let A be algebraically compact and A 0B = C a direct product of cyclic p-groups [see (38.2)]. Collect the summands Z(p'') belonging to the samep and form their direct product C, . This is a subgroup of C, and clearly, C= C, . The C, are fiilly invariant in C, hence (9.3) implies C, = A, @ B, where A, = A n C,and B, = B n C, . Thus, C = A, 0 B, . Obviously, the A, with varyingp generate their direct sum A, in A . If C is regarded as a topological group in the Z-adic topology, then it is clear that the closure of A , in C must contain A, hccause of the divisibility of A P I A , . Since B' = 0, A is closed in C, and therefore we obtain the inclusion A, 5 A . Analogously, B, 5 B, and consequently, A = A, and B = B, . As a direct summand of a complete group, A , must be complete in its Z-adic topology, which is now identical with the p-adic topology, for the Z-adic and p-adic topoIogies coincide on Z( p'). Conversely, let A , be a group complete in its p-adic topology. The following definition makes A , in a natural way into a module over the ring Q*,of the p-adic integers. Let a E A, and n = so + s,p + ... + snpn+ * - . E a*,.The sequence sea, (so s,p)a. . . ., (so + sIp + ... + snpn)a, ...

fl,

n,

n,

n, n, n, n, n, n,

+

is Cauchy in A , , hence it tends to a limit in A, which we define as na. It is straightforward to show that, in this way, A , has actually become a uiiital

VII.

168

ALGEBRAICALLY COMPACT GROUPS

QS-module. Hence we conclude qA, = A , for all primes q # p, that is, the Z-adic and p-adic topologies of A , are the same. By (39.1), A , is algebraically compact, and (38.3) implies the algebraic compactness of their direct product A . Finally, the uniqueness of the components A , in (1) follow at once from the relation A,

=

n

( k = i , 2 , -1,

4kA

9fP

where q again denotes primes. This is a consequence of the relations qA, (4 # PI and n k PkA, = 0.0

=A,

The group A , of the preceding theorem will be called the p-adic component of A . It is a p-adic algebraically compact group in the sense that it is complete in its p-adic topology. As is shown by the preceding proof, it must be a p-adic module. It can be characterized completely via basic subgroups, as is indicated by the following theorem.

Theorem 40.2. There is a one-to-one correspondence between the p-adic algebraically compact groups A , and direct sums B, of cyclic p-adic modules: given A , , we let its basic submodule B, correspond to A,; to a given B, , there corresponds its p-adic completion. It follows immediately from (35.5) that the correspondence A,- B, with A , a p-adic module and B, its basic submodule is single-valued [B, is now regarded as an abstract group]. On the other hand, the completion B of a direct sum B of cyclic p-adic modules, with respect to the p-adic topology, does exist and is unique up to isomorphism; B is algebraically compact [cf. (39.1)]. If we identify B with a subgroup of B under the natural embedding, then it is sufficient to refer simply to (39.6) in order to conclude that B is basic in 8. Therefore, we infer that if the correspondence B,H A, is followed by the correspondence A , w B , , then we get back to the original direct sum of cyclic p-adic modules. We still need to verify the isomorphy of two p-adic algebraically compact groups A , and A ; under the hypothesis of having isomorphic basic submodules B, . We start with the diagram

0 -+ Bp-+ A , I

-+

A,/B,

-+

0

/

where the row is pure-exact and p is the injection B, A ; . By the pureinjectivity of A ; , there is a homomorphism 4 : A , A ; making the diagram commutative. Analogously, we get a homomorphism $ : AI, --* A , that extends the injection map B, -+ A , . Hence $4 : A , -+ A , , whose restriction --f

-+

40.

169

THE STRUCTURE OF ALGEBRAICALLY COMPACT GROUPS

to B, is the identity, thus $4 map of Aj, whence A , Aj,

=

.n

lA, by (34.1). Similarly,

41)is the identity

In view of the foregoing theorem, the p-adic algebraically compact groups A , can be characterized by the same systems of invariants as their basic submodules B, , namely, by the cardinal numbers in, and mk (k = 1,2, .) of the sets of components [in a direct decomposition of B,] isomorphic to J , and t o Z(pk),respectively. This countable system of cardinals is a complete, independent system of invariants for A , , from which we can reconstruct A , by forming first the obvious direct sum and then p-adic completion:

A , E p-adic completion of

[@ J , 0 mo

m

@ @ Z ( p k ) ]. k = l mk

If A is an arbitrary algebraically compact group, then the invariants of its maximal divisible subgroup along with the invariants of its p-adic components A , constitute a complete, independent system of invariants for A . We are therefore justified to claim that the structure of algebraically compact groups is known. Example I. Let in be a n infinite cardinal. Then J,” p-adic completion of @ J, . 2m

For, it is a torsion-free p-adic algebraically compact group whose basic submodule is of the same rank as J,”/pJ,” g Z ( P ) ~ . Example 2. Let

mk

(k = 1, 2, . ..) be any cardinals. Then

fi [2 Z(pk)]

nmk.

E p-adic completion of

k=1

[@m

Jp

0

& @ Z(pk)]

k=lmk

where m = In fact, the group A in question is p-adic algebraically compact whose basic submodule has the indicated torsion part, by (33.3). Clearly, A/pA 2 @,,Z(p) and the torsion subgroup of A is represented by the elements of the direct sum in this direct product. From the proof of (35.2), we get m

nk

=nm,.

In view of the above theorem, we are now able to derive the following elementary, useful corollaries. Corollary 40.3. r f a reduced algebraically compact group is torsion, then it is bounded. In a decomposition (1) of such a group A , only a finite number of A , # 0 may occur. By (40.2), each A , is the p-adic completion of a direct sum B, of cyclic p-groups. It is straightforward to check that this completion contains elements of infinite order, unless B, is bounded when Bp = AP.u Corollary 40.4 (Kaplansky [2]). Every reduced algebraically compact group # 0 contains a direct summand isomorphic to J , or Z(pk)(k = 1, 2, . *) for some p .

170

VII.

ALGEBRAICALLY COMPACT GROUPS

If the group A # 0 is algebraically compact and reduced, and if p is a prime such that A , # 0 [see (40.1)], then the basic submodule B, of A , is not 0, and hence has a direct summand of the indicated type. This is pure in A and algebraically compact, thus it is a summand of A , t o o . 0 It follows that the directly indecomposable algebraically compact groups are: J,, Q, and subgroups of Z(p"). [That J , is indecomposable will also be shown in (Chapter XIII).]

EXERCISES 1. Every reduced algebraically compact group is a unital module over the ring a,*. 2. If, foreach prime p , A , is a group complete in its p-adic topology, then the product and box topologies coincide in A = A,. 3. (a) A countable algebraically compact group is the direct sum of a divisible and a bounded group. (b) There exist no countable reduced algebraically compact torsion-free groups. 4. Determine the invariants of the following algebraically compact group :

n,

n,

[

kDIZ(pk)]m 0 J):

(in, ti infinite cardinals).

5 . If A and A' are algebraically compact groups, each isomorphic to a pure subgroup of the other, then A z A'. 6 . If A is an algebraically compact group such that A @ J , z A' 0 J , , then A A'. 7. If A is algebraically compact and A 0 A E A' 0 A', then A E A ' . 8. (Mader [I]) A reduced algebraically compact group A is the 2-adic

completion of a direct sum of cyclic groups exactly if the rank of A / [ p A T(A)] is independent of the prime p.

+

41. PURE-ESSENTIAL EXTENSIONS

The striking analogy between divisible and algebraically compact groups can be pushed further by pointing out the analog of the divisible hull. We first introduce the following concept. Let G be a pure subgroup of A , and let K(G, A ) denote the set of all subgroups H 5 A such that (i) G n H

=0

and

(ii) (G

+ H ) / H is pure in A / H .

Since G + H is the direct sum of G and H , condition (ii) amounts to the fact that if nx = g + h (g E G , h E H ) is solvable in A , then there exists a solution of ny = g in G. Hence K(G, A ) contains, along with H , all subgroups of H. The assumed purity of G in A implies 0 E K(G , A ) , i.e., K(G, A ) is not empty.

41.

171

PURE-ESSENTIAL EXTENSIONS

Following Maranda [l], we call the group A a pure-essential extension of its subgroup G if G is pure in /1, and if K(G, A ) consists of 0 only. That is to say, if G is pure in A and if 6 is a homomorphism of A inducing a monomorphism on G such that 4 G is pure in 4.4, then Q, is a monomorphism. The analogy with essential extensions is apparent. Let us begin by introducing some lemmas. Lemma 41.1 (Maranda [ 11). If C is a pure subgroup of A , then there exists a homomorphism 4 of A which induces a monomorphism on G and f o r Ichich 4 A is a pure-essential extension of 4 G . The set K(G, A ) of subgroups H of A is inductive. In fact, if is a chain in K(G, A ) , and if H is the union of this chain, then (i) is obvious, and if n x = g + h ( g E G , h E H ) is solvable in A , then for some i, 12 E H iE K(G, A ) , whence ny = g is solvable in G, and H E K(G, A ) . By Zorn’s lemma, K(G, A ) contains a maximal member B. Define 4 : A A / B as the canonical epimorphism; then 4 I G is a monomorphism, and 4 G = (G + B ) / B is pure in 4 A = A / B . The maximal choice of B implies K(q5G, + A ) = {O}.o --f

Lemma 41.2. Let A i (i E I ) be a chain of groups and A their union. If each

A iis a pure-essential extension of G , then A has the same property.

It is immediately clear that if G is pure in every A i , then it is pure in A , too. If some H # 0 belonged t o K(G, A ) , then for some i, H n A i # 0, and one would have H n A iE K(G, Ai),a contradiction.0 Lemma 41.3 (Maranda [l]). If C is Q pure-essential extension of G , and i f A is an algebraically compact group containing G as a pure subgroup, then the identity map of G can be extended to a rnonomorphism of C into A . Since 0 + G + C is pure-exact and A is pure-injective, there is a homomorphism 4 : C - t A which induces the identity map on C . Since G is pure in A and hence in 4C, 4 must be a monomorphism.0 The group A will be called a maximalpure-essential extension of G if it is a pure-essential extension of G, and if A‘ with A < A’ is never a pure-essential extension of G. The existence of such maximal extensions may easily be established : Proposition 41.4 (Maranda [ 11). Erery pure-essential extension of a group G is contained in a maximal pure-essential extension of G. Let C, be a pure-essential extension of G. By way of contradiction, suppose that G has no maximal pure-essential extension containing C,. On account of (41.2), this means that, for every ordinal (T, there is a well-ordered, properly ascending chain of groups

c, < c, < ... < c,

172

VII.

ALGEBRAICALLY COMPACT GROUPS

such that for every p 5 6,C, is a pure-essential extension of G. Let A be a n algebraically compact group containing G as a pure subgroup. (41.3) guarantees the existence of a monomorphism C, + A , which is clearly impossible if 0 is of larger cardinality than A . 0 The maximal pure-essential extensions can easily be described :

Proposition 41.5. A maximal pure-essential extension of a group G is a minimal algebraically compact group containing G as a pure subgroup. Let A be a maximal pure-essentialextension of G and B a group containing A as a pure subgroup. By (41.1), there is a homomorphism 4 of B which induces a monomorphism on G such that 4 B is a pure-essential extension of 4G. Hence 4 G is pure in + A , and the definition of pure-essential extensions implies that 4 is monic on A , too, so 4 A z A is a maximal pure-essential extension of 4 G E G. Thus 4 B = &A, and 4 followed by the inverse of 4 1 A is a projection of B onto A . Hence A is a direct summand of B, and A is algebraically compact. If A’ is an algebraically compact group such that G 5 A’ A , then by (41.3) there exists a monomorphism $ : A + A ’ which induces the identity on G. Obviously, $ A is again a maximal pure-essential extension of G, and therefore $ A 5 A’ 5 A implies $ A = A.’ Hence A’ = A , and A is minima1.O

s

Proposition 41.6. A minimal algebraically compact group containing G as a pure subgroup is a maximal pure-essential extension of G. If A is a minimal algebraically compact group containing G as a pure subgroup, and B is a maximal pure-essential extension of G, then, in view of (41.3), there is a monomorphism 4 : B -+ A which is the identity on G. Evidently, 4 B is, by (41.9, an algebraically compact group containing G , hence $B = A because of the minimality of A . 0 Now we have arrived at the main result of this section.

Theorem 41.7 (Maranda [I]). For every group G, there exists a minimal algebraically compact group A that contains G a s apure subgroup. A is a maximal pure-essential extension of G and is unique up to isomorphism over G . Given G, by (41.4) there is a maximal pure-essential extension A of G. (41.5) and (41.6) imply the first two assertions. Let both A and A‘ be minimal algebraically compact groups containing G as a pure subgroup. The identity map of G can be extended, owing to (41.3), to a monomorphism 4 : A + A ’ . Here 4 A is minimal algebraically compact containing 4 G = G as a pure subgroup, whence $ A = A ‘ , and 4 is an isomorphism.u We may call a minimal algebraically compact group containing a given group G as a pure subgroup the pure-injective hull of G. The last part of the preceding proof shows that every algebraically compact group containing

41.

PURE-ESSENTIAL EXTENSIONS

173

G as apure subgroup contains a pure-injective hull of G. The following result provides a useful criterion. Lemma 41.8. An algebraically compact group A containing C as a pure subgroup is the pure-injectire hull of G exactly i f the following conditions hold: (i) the maximal divisible subgroup D of A is the injective hull of G' ; (ii) the quotient group A / G is ciiiisible. First, assume A is the pure-injective hull of G . Write A divisible and C reduced. I n view of the purity of G in A ,

c1= nnc = n ( G n n A ) = G n A '

=

D @ C with D

= G n D.

If D were not the divisible hull of GI, then we could write D = D,@ D, with G' 5 D, and D, # 0. Here D, E K ( G , A ) , which is absurd. If A/G were not divisible, then let E/G = (A/G)' < A/G. Clearly, D 5 E, thus E = D @ ( C n E ) . From C/(C n E ) E (C + E ) / E = A / E E (A/G)/(E/G),it follows that C n E is closed in the complete group C, thus is itself complete. Hence E would be algebraically compact, properly contained in A . This proves (ii). To prove sufficiency, assume (i) and (ii). By what has been noticed before the lemma, there is a pure-injective hull E of G in A , G 5 E 5 A . From the first part of the proof we know that E/G is divisible, whence E is pure in A , so E is a direct summand of A , A = E @ E, . By (ii), E, is divisible, E, 5 D, and therefore E, n G = 0 and (i) imply E, = 0 . 0 Now it is easy to characterize the pure-injective hull of a given group G. Theorem 41.9. The pure-injectice hull of a group G is isoniorphic to the direct sum of the injectii)e hull of G' and the completion of G. Let D be the injective hull of GI, and let 4 : G D extend the inclusion map GI + D. If p : G + G is the canonical map, then the homomorphism (4 @ p)A : G --t D @ G is obviously monic, and the purity of pG in G implies Im (4 @ p)A pure in D @ G. Combining (39.5) with (41.8), the assertion is c1ear.a --f

Recall that the divisible hull of GI is determined by the ranks .of GI, while the completion G is determined by the p-basic subgroups of G. Hence, these invariants serve to characterize the pure-injective hull of G.

EXERCISES Let G be pure in A and let H E K(G, A ) . Then K E K(G, A ) , if K / H E K((G + H ) / H , AIH). 2. Prove that H E K(G, A), if H is contained in the first Ulm subgroup A' of A and H n G = 0. 1.

VII.

174

ALGEBRAICALLY COMPACT GROUPS

3. Let A be a p-group and B a basic subgroup of A . Then K(B, A ) is the set of all subgroups of A ' . 4. Assume C is an algebraically compact group containing G as a pure subgroup. Prove that the pure-injective hull A of G in C is a direct summand of C. 5. Let B be a pure-essential extension of C and A a pure-injective hull of B. Prove that A is a pure-injective hull of C, too. [Hint: A contains a pure-injective hull A' of C such that B 5 A ' ; A = A ' @ A" with A" divisible; from (41.8), conclude A" = 0.1 6 . If B is a pure-essential extension of C, then BjC is divisible. [Hint: (41.8) and Ex. 5.1 7. A pure-essential extension of a pure-essential extension is a pureessential extension. [Hint: Ex. 5.1 8. Let C be pure in B and B pure in A , such that A is a pure-essential extension of C. Then B is a pure-essential extension of C and A is a pure-essential extension of B. 9. Characterize the pure-injective hull of the example in 35. 10. If A i is the pure-injective hull of G i(i E I ) , then A i is not always the pure-injective hull of G i.

fl

42.*

MORE ABOUT ALGEBRAICALLY COMPACT GROUPS

In our subsequent developments, we shall frequently meet algebraically compact groups; our knowledge on their structure will enable us to get a better insight into the structure of certain groups. There are some results on algebraically compact groups which d o not fit properly into our future discussions; since they are worthwhile noticing, we intend to discuss them in this section. The first result will show that algebraically compact groups are abundant among quotient groups of direct products. Let G i( i E I ) be a family of groups with an arbitrary index set I, and let K be an ideal in the Boolean algebra I of all subset of I . Furthermore, let K* be the a-ideal generated by K , i.e., it consists of all subsets of unions of countable many members of K [it is easy to see that these members of K may be assumed to be mutually disjoint]. We are concerned with the K- and K*-direct sums of the Gi: Theorem 42.1 (Fuchs [ I I]). If K and K* are dejned as abore, then the quotient group (1)

is algebraically compact.

A =

OK* Gi/ OKG i

42.

MORE ABOUT ALGEBRAICALLY COMPACT GROUPS

175

Ignoring a trivial case, we suppose K # K*. From (38.5) and (38.1), it follows that it suffices t o establish a solution in A for the system k

with a countable number of unknowns and equations, under the hypothesis that every finite subsystem of ( 2 ) admits a solution in A. We first pick out representatives a j E i i j , where a, E O K *G j , and focus our attention on the index set

5=

u OCI

s(aj);

j= 1

since all the supports s(aj) belong to K*, manifestly 5 E K*. If 5 E K, then xk = 0 is a solution. Therefore, suppose 5 -$ K . By hypothesis, the subsystem (2,) of (2) consisting of the first m equations of (2) is solvable in A ; assigning 0 to the unknowns not occurring i n ( 2 , ) explicitly, we obtain values xk = g(k, m ) E A for every m , which satisfy the first m equations in (2). We proceed to the system

C

(3)

k

njkxk

=Uj

(Uj

E

OK* Gj ;

j = 1, 29 "'),

and select some representatives g(k, m ) E g(k, rn) in OK,G i ,with 0 as a representative of 0. The system ( 3 n 1 )consisting of the first m equations of (3) need not be satisfied by xk = g ( k , m ) , but the set 5 , of indices i E Z for which the ith components of x k njkg(k, m ) and a i [as elements of OK*G i ]are different for j = 1 , 2, . . . , m is certainly a member of K. Starting with the sets rl, ... , t,, ... E K, we define successively the sets lo, i l , . * ., i,, ... E K so as to be pairwise disjoint and to satisfy i o = 51,

i 1

3

52\io>

e 3 . 3

i m

2

5m+l\(io

u ... u i m - l ) ?

* * .

and, in addition,

u m

[,=

< u (, u . . . u therefore by (1 1.l) there exists a unique (5)

(i S j ; k 5 I),

0 : H + Hom(A

I D, C I I) such that

HOm(pk, x i ) = d d i k .

Again by (44.4), the left side map is monic. The same must hold for 6,for if h E H belongs to Ker r ~ ,then h = d k i qki [with some u k i E Hom(A/Ak, C,)] Satisfies HOm(pk, ni) t,7ki = d 0 k i I]ki = dh = 0, thus q k i = 0 and h = 0. TO prove cr epic, take into account that the groups Hom(p,, x i ) together exhaust Hom(A I D, C I I) whence (5) guarantees that Q is an epimorphism, and consequently, an isomorphism. It is clear from the definition that it is natural.0

EXERCISES 1. Check what was stated for I- and D-exactness of sequencesin the mentioned examples. 2. (Fuchs [12]) For a category (d, I), let Hom,(A, C ) be the set of all morphisms ( A , I,) + (C, Ic). Prove that: (a) Hom,(A, C) is a subgroup of Hom(A, C ) ; (b) it is a left-exact functor in both variables. 3. (Fuchs [12]) For a category ( d ,D), let Hom,(A, C) mean the set of all morphisms ( A , D,) + (C, Dc). Verify the analogs of (a), (b) in Ex. 2. 4. Let ( d ,I) be defined such that 1, consists of all subgroups of A which belong to a class X of groups [closed under taking direct sums, subgroups and epimorphisms], and assume ( d ,D) is the category with D A consisting of all A’ 5 A with AIA’ E 3. Show that Hom(A I D, C) = Hom(A, C I I). I) and ( d ,D) where Hom(A 1 D, C I I) commutes 5. Give examples for (d, with infinite direct sums in both variables. [ H i n t : subgroups of finite order and of finite index, respectively.] 6. Let (a, I) be the category with I, the bounded subgroups of A . If B is a basic subgroup of a p-group A , then : (a) Hom(A, C I I) z Hom(B, C I I); (b) Hom(B, C I I) is the torsion part of Hom(B, C); (c) if A = @ A i , then Hom(A, ClI) is the torsion part of the direct product of the Hom(Ai, C I I).

46.

193

HOMOMORPHISM GROUPS OF TORSION GROUPS

7. Let ( d ,I) be the category with I, the finite subgroups of A . Let A be a torsion or a complete group, and A , the p-component of A . Prove the isomorphism Hom(A, C I I) z @ Hom(Ap, C). P

8*. (Pierce [I]) Let Hom,(A, C) denote the group ofall small homomorphisms of the p-group A into C [for definition, see next section]. Prove that the functor Horn,(*, C) carries a pure-exact sequence into an exact sequence. 46. HOMOMORPHISM GROUPS OF TORSION GROUPS

Having considered some elementary properties of Hom and the exact sequences for Hom, we may turn our attention to find the structure of Hom(A, C) in certain cases. Evidently, Hom(A, C) depends on the given groups A and C, and Hom (Z, C) z C shows that every group C occurs as Horn. It is, however, a remarkable fact that, if A is supposed to be a torsioh group, then Hom(A, C) must be algebraically compact, and hence its characterization in terms of invariants of A and C can be hoped for. Because of (43.4), only p-groups are to be dealt with. Theorem 46.1 (Fuchs [9], Harrison [2]). If A is a torsion group, then Hom(A, C ) is a reduced algebraically compact group, for any C. First proof. We prove that if A is a p-group, then Hom(A, C ) is complete in its p-adic topology [cf. (40.1)]. To prove H = Hom(A, C) is Hausdorff, let q E H be divisible by every power of p . Given a E A , say of order pk, let x E H satisfy p k =~q. Then qa = pkXa = Xpka = 0 implies q = 0. Next let ql, q,,, be a Cauchy sequence in H ; dropping to a subsequence if necessary, we may assume it neat: v,,+~ - q,, E ~ " Hfor every n, that is, q,,+ - q,,= pnxnfor some x,,E H . Define 5 s . )

v = v, + (?*

- Vl)

+

* * -

+ ( ? n + l - v.) + " ' ;

this is a homomorphism A -, C, since for a E A of order pk, ( I , , + ~- q,)a = 0 for all n 2 k , showing that qa = ?,a (q2 - q l ) a * . * (qk - q k - & is well defined. Furthermore,

+

I1-q

n=(qn+l-Iln)+(qn+Z-Iln+l)+...=Pn(Xn+PXn+l

+ +

+***)

where, again, I,,+ p ~ , , + + ~ belongs to H , i.e., q - q,,E ~ " H and q is the limit of the given Cauchy sequence. Consequently, H is complete. Second proof. A , as a torsion group, is the direct limit of its finite subgroups A i . By (44.2), Hom(A, C) is then the inverse limit of the groups

VIII.

194

HOMOMORPHISM GROUPS

Hom(A, , C) which are bounded in view of (43.3). Hence, Hom(A, C) is the inverse limit of reduced algebraically compact groups, therefore the assertion follows from (39.4).0

In order to find out the exact structure of Hom ( A , C) for torsion A , we may restrict ourselves, without loss of generality, to the case when both A and C are p-groups. Let cc

B = @ B,

(1)

with

B,

n= 1

=

@ Z(p") m,

be a basic subgroup of A , and assume C reduced. The exact sequence 0 .+ B % A --t A / B + 0 induces the exact sequence 0 = Hom(A/B, C) -+Hom(A, C)mf_+ Hom(B, C) showing that Hom(A, C ) can be viewed as a subgroup of Hom(B, C). The two groups have the same torsion subgroup. In fact, i f q E Hom(B, C) satisfies pkq = 0, then pkB5 Ker q and q maps B/pkB into C. If the homomorphism of A/pkctB= ctB/pkctB@G (with G A/rB) which acts as q on the first component and trivially on the second component is viewed as a homomorphism of A into C, then X C I = q and q E Im c1*. From (46.1) it follows that Hom(A, C) contains the p-adic completion of its torsion subgroup which is necessarily a summand of Hom(A, C ) . Its complement must be a p-adic algebraically compact torsion-free group.

=

From (43.3) we infer Hom(B, C) =

fi fl

n = l m,

C[pn].

The groups C [ p n ]are direct sums of cyclic groups of orders Sp", and the same holds for f l , , C [ p " ] . The invariants of these groups can be evaluated by means of cardinal invariants of C, thus (33.3) can be applied to describe the basic subgroup V of the torsion part of Hom(B, C). Therefore, Hom(A, C) is the direct sum of the p-adic completion P of V and a p-adic algebraically compact torsion-free group which will be known as soon as its basic subgroup W is determined. In the proof of the key lemma (46.3), the following set-theoretic lemma will be made use of. Lemma 46.2 (Pierce [l]). Let I be a set of infinite cardinality in, and n a cardinal number such that 0 < n 5 in. Then there exists a set { I j } j s J of subsets of I such that (i) lIil

= n for

every j ;

46.

HOMOMORPHISM GROUPS OF TORSION GROUPS

195

(ii) IJI = in"; (iii) the I j are independent in the sense that, $ I o , 11,. ' . , I, are distinct elements of the set, then I , is not contained in the union Il u * .. u I , . The following simple proof is due to P. Erdos and A. Hajnal. Let K be a s e t of cardinality i n : we can decompose it into n disjoint subsets L , , each of cardinality 111. Clearly, if {Kj}jEJ is the family of all subsets of K which contain exactly one element from each L , , then this family satisfies (i) and (ii), while (iii) is replaced by the condition that no Kjcontains another one. Now define Ij as the set of all finite subsets, of K j . Then the set { I , } j s J satisfies not only (i) and (ii), but (iii) too, since if ciE Io\Ii for i = 1 , ... , n, then {cl, ... , c,} E I , but 4 Il u ... u I , . The proof will be completed by identifying the elements of the given set I with the finite subsets of K, under a one-to-one correspondence.0 Now we are ready t o prove:

Lemma 46.3 (Pierce [ I ] ) . If A , C are reduced p-groups whose basic subgroups B , D are of inJnite$nal ranks nt, n, then the basic subgroup W of X is of rank nm. An inference like the one at the beginning of the proof of (36.1) shows that bounded direct summands can be separated from A and C to achieve r(B) = in and r ( D ) = it. Bounded direct summands have no influence on W , so the assumption r ( B ) = 111, r ( D ) = 11 means no loss of generality. From (34.4) we get the estimation J C JI I D I K awhence IWI 5 IHom(A, C)l

5 IHom(B, C)i 5 ICIIBi5 (DIKalBI= n"'

Consequently, it suffices to prove the existence of a p-independent set S of elements of infinite order in Hom(A, C) such that IS1 = it'". The main idea of the proof is to concentrate on small homomorphisms 4 : A + C which we define by the following condition: (*) for every k 2 0 there exists an nJk) such that e(a) 2 n+(k) implies e ( 4 a ) 5 e(a) - k .

These have the advantage that they are completely determined by their restrictions to B and can be chosen arbitrarily on B. In fact, if 4 is a small homomorphism of A into C, and if a E A is of order pk, then choose n = n+(k) according to (*) and write a = p"g + b with g E A , b E B such that e(p"g) 5 e(a) [which can be done, B being pure i n A ] . Then e ( 4 g ) 5 e ( g ) - k implies p"4g = 0, and so &z = 4b. Furthermore, if 4'is a small homomorphism of B into C, then putting $a = 4'b we obtain a small homomorphism 4 : A .+ C [with n+(k)= n+.(k)]. Therefore, small homomorphisms of A and B are essentially the same.

VIII.

196

HOMOMORPHISM GROUPS

It is routine t o verify that the small homomorphisms of A into C form a subgroup H of Hom(A, C ) . The quotient group Hom(A, C)IH is torsion-free, for if q E Hom(A, C) satisfies prq E H , then together with prq, q also satisfies (*) with n,(k) = n,,,(k + r ) . Hence H is pure in Hom(A, C), and it is therefore enough to find an S of the desired nature in the subgroup H . We distinguish two cases.

Case I : n

5 m. Choose a direct summand DO

=

6

n=l

of D such that 0 < e(dl) < .. . < e(d,,) < * way : B

(2)

m

m

= @

- - , and decompose B in the following

G,

where

G,,= Q ( b n i ) i G l n

n=O

and fin r(G,,) = 11 1 . = m, e(bni)2 2e(d,,)2 2n for n 2 1 ; the index sets I,, (n = 0, 1, 2, * .) are of course disjoint. In Z,,we select subsets I,,, ( j E J ) such that

(3)

IJI = nm = mm,

IZnjl = m,

Znj are independent;

and

this can be done by (46.2). There is no loss of generality in choosing the same J for every n. Let us define homomorphisms 4 j : B + Do as follows:

4 j bni=

("

if i E Z n j , n L 1 , otherwise.

0

These 4 j are small homomorphisms, for e(d,,)5 +e(b,,J,i.e., e(bni)2 2k implies e ( 4 jbni)5 e(bni)- k , and hence it is easy to conclude that (*) holds with n,(k) = 2k. Every 4 j is of infinite order, because d,, E Im 4 j for all n. The 4 j are p-independent, for if

4=m,4o+rn,4l

+*-*+ms4,Ep'H

(rn,#O),

then by (3), there is an index i, E I,,, that does not belong to the union *. u Z,, . For this i, , 4obnio= d,,, while +,bni,,= * . * = 4, bnio= 0 , thus 4bnio= m, d,, . This element must belong top'C, whence the purity of Do implies pr I ma . Analogously, pr I m,, . . . , rn, , proving that the set S = { 4 j } is p-independent, of cardinality nm.

Z,, u

-

Case ZZ: n > m. This time we decompose D as follows: m

D

(4)

=

Q GA

where

n= 1

where lK,,l (5)

=n

G;

=

@ (dnk),

keK.

and e(dnk)2 n, and choose subsets K,,, ( I E L ) in K,,such that

IKnJ= m,

ILI

= nm,

and K,,,are independent.

46.

197

HOMOMORPHISM GROUPS OF TORSION GROUPS

Letf,, be a mapping of K,, into I,, of (2), such that each i E I, is the image of at most one k E K,, , and iff,l k = i , then 2e(d,,) 2 e(bni).It is clear that such an f , , does exist for all n, 1. For every I E L we define a homomorphism $ 1: B --* D by putting 'lbni

=

{tk

if i =f , , k , otherwise.

As above, it follows that is a small homomorphism of infinite order. In order to verify thep-independence of the set S = { $ l } l E L , let

$ = molClo

+ ml$l + + m,$,EprH

( m , z 0).

..*

Some k, E K,, exists such that k , $ K,, u ... u K n S For . i, =Lok, we have $, bnio= dnko,while $,bnio= * * * = $s bnio= 0. From $bnio= mo dnkowe infer m , . This establishes the as in Case Z that pr I m, , and analogously, pr I m,, p-independence of S . 0 a ,

Now we are in a position to prove the main result. We shall use the following notations. Let (1) denote a basic subgroup of A and m

(6)

D

=

@ D, n=

with D,

1

=

@ Z(p") nn

a basic subgroup of C. Let us put m

(7)

= fin r ( B ) ,

n

fin r(D),

=

p

= fin r(C).

We define, for cardinal numbers u, D,

if u is finite, if u is infinite,

d(u>0) = (y;D)u

and notice that

[0Z(P")I"

=

I)

0 Z(P"1.

d(u,o)

Theorem46.4 (Pierce [l]). Let A and C be reduced p-groups. Then Hom(A, C) is a p-adic algebraically compact group whose basic submodule is

where for n

>=

1

and

ro = d (m,n).

WII.

198

HOMOMORPHISM GROUPS

It is clear that C[p"] = D,@ @ D,-,@ En,where En is a direct sum o f - p + ~ ~ = , ,copies n , of Z(p"). Since r, is the cardinal number of the set of direct summands Z(p") in a basic subgroup of the torsion part of the group C [ p " ] ,as is shown by (33.3), we infer

n:=

m

with p, = ~ k m , , + l i n , .This proves the formula for r, (n 2 l), while that for ro is the content of (46.3), where it is trivial for vanishing m or n . 0 In view of this theorem, the algebraic structure of Hom(A, C) can completely be described if A is a torsion group and C is reduced. If C happens to contain subgroups Z(p"), then (46.4) and 47, Exs. 8-10 together yield the structure of Hom(A, C).

EXERCISES 1.

(Pierce [l]) Show that condition (*) is equivalent to the following one: for every k 2 0 there exists an n such that e(a) 5 k and h(a) 2 n imply $a = 0. 2. (a) If $ is a small homomorphism of A , then B

3. 4.

5.

6. 7.

+ Ker $ = A

for any basic subgroup B of A . (b) A' 5 Ker $. Let A = B @ D with B bounded and D divisible. Describe the small homomorphisms of A into any group. (Pierce [l]) (a) In each of the following cases there is a homomorphism of A into C which is not small: (i) A is unbounded and C is not reduced; (ii) A is unbounded and C contains a subgroup = A ; (iii) A has an unbounded basic subgroup and is countable, while C is not bounded. (b) All homomorphisms of A into C are small if: (i) A has bounded basic subgroup and C is reduced; (ii) C is bounded. (Pierce [l]) Let B = @ipl(ai)be a basic subgroup of A , and let c iE C (i E I ) satisfy: (i) e(ci) 5 e(a,); (ii) for any k 2 0 there is an n such that e(ci)2 n implies e(ci) 5 e(ai)- k . Then there is a unique small homomorphism $ of A into C such that $(ai)= c i (i E I ) . (Pierce [l]) Let G be a pure subgroup of the p-group A . Every small homomorphism of G into C can be extended to one of A into C. [Hint: Ex. 5.1 (Pierce [l]) (a) $ is a small homomorphism if pm$ is small for some m . (b) A small homomorphism of p m A into pmC can be extended to a small homomorphism of A into C. [Hint:Ex. 5.1

CHARACTER GROUPS

47.

199

8. Let A , C be as in (46.4). Determine the final ranks of Hom(A, C) and its basic subgroup. 9. Let C be acotorsionp-adicmodule(see54), and let A have torsion p basic sub group. Then Hom(A, C) is algebraically compact. 47. CHARACTER GROUPS

In the preceding section we examined Hom(A, C) for torsion groups A and reduced groups C. Our next aim is to investigate the case when A is arbitrary and C is the additive group K of the real numbers mod I , in which case Hom(A, K)-equipped with a suitable topology-is known as the character group Char A of A . I n this section, we concentrate on the algebraic structure of Char A ; the fact that it carries a topology will be irrelevant in most of our results in 47. Our results will give complete information about the structure of character groups. Algebraically, K is nothing else than the direct product of quasicyclic groups, one for each prime p ; hence Char A

z

n Hom(A, Z(p")). P

Consequently, it suffices to deal with Hom(A, Z(p")). Let a

B = @ B,

where

B,

=

@ Z , B, = @ Z(p")

for n 2 1,

mn

mo

fl=O

be ap-basic subgroup of A . Thep-component of A / B isof theform B m Z ( p m ) ; the fact that 111 is not an invariant is inessential, but it can be made unique by choosing a lower basic subgroup in the p-component of A . Finally, let f denote the torsion-free rank of A/B. [These cardinals will be denoted by in&), . * . , t(p), if their dependence on p cannot be suppressed.] The following theorem gives complete information about the homomorphism groups Hom(A, Z(p")). Theorem 47.1 (Fuchs [S]). For any group A , (1)

Hom(A, Z W ) )

n in o

nn

Z(pZ) 0

n= 1

Illll

Z(p")0

n Ill

J,

0 11Q. fNo

Owing to (44.5) and (44.7), the p-pure exact sequence O + B + A / B 0 implies the p-pure exactness of the induced sequence 0 + Hom(A/B, Z(p")) -+ Hom(A, Z ( p " ) ) + Hom(B, Z(p")) -+ 0. By virtue of (43. I ) A

+

--f

m

a

200

VIII.

If we write A/B = @,Z(p")

@ G,

Horn( 0 Z(p"), Z(p-1) m

=

HOMOMORPHISM GROUPS

with G[p] = 0, then because of

n Hom(Z(p"), Z(p"N m

=

m

J,

,

it remains to evaluate Hom(G, Z(p")). Let H be the p-pure subgroup in G generated by a maximal independent system of elements of infinite order in G; this H is well defined, since in G division by p is unique, and evidently, Hz Q(,)such that G/H is a torsion group with 0 p-component. The exactness of 0 -+ H -+ G -P G/H + 0 implies the exactness of the sequence 0 = Hom(G/H, Z(p")) Hom(G, Z(p")) -+ Hom(H, Z(p")) -+ 0, thus --f

where we have used example 4 in 43. We see that Hom(A/B, Z(p")) is the direct sum of a divisible group and a p-adic algebraically compact group, so its p-purity in Hom(A, Z(p")) implies that it is a direct summand of Horn@, Z(p")).CI Notice that the group in (1) may be given a more explicit form by making use of examples in 23 and 40. If we determine the cardinal numbers m,, m,, m, I for every prime p, then Char A can be determined as the direct product of the groups (l), with p ranging over all primes p. Notice that the first and fourth summands in (1) arise from elements of infinite order, while the two middle summands from elements of p-power orders. Hence Corollary 47.2. Char A is reduced if and only is divisible if and only i f A is torsion$ree.O

if A is a torsion group, and

A glance at the group ( 1 ) shows that it is algebraically compact. This yields the known fact that the character groups are algebraically compact, without making use of the deep theorem that the character groups of discrete abelian groups [with the suitable topology] are just the compact abelian groups.

It is natural to inquire about conditions under which an algebraically compact group can carry a compact topology, that is to say, is a character group. The conditions are not difficult to state: they are inequalities between cardinal numbers [cf. Ex. 51. We prefer to formulate the relevant conditions in the simpler forms given in the two subsequent corollaries. Clearly, the cardinal numbers m, and m can be chosen arbitrarily and independently for every p ; we thus have: Corollary 47.3 (Hulanicki [2], Harrison [l]). A reducedgroup is the character group of some (torsion) group exactly if it is a direct product of cyclic p adic modules [ p need not beJixed1.n

47.

20 1

CHARACTER GROUPS

For divisible groups, a simple inequality must be satisfied: Corollary 47.4 (Hulanicki [ I ] , Harrison [I]). A divisible group # 0 is the charactergroup of some (torsiongree) group i f , and only if, it is of the form

nn p

Z(pm)0

rP

n r

Q

where r

1No.

If A is torsion-free, then its rank is, in the above notation, tn,(p) + €(p), showing that Char A will have the indicated form with rp = mo(p), unless f(p) = 0 for every p , in which case, however, the direct sum with Q does not change the structure of the first direct product. Conversely, given a divisible group of the above form, a simple calculation shows that the group structure does not change if r is replaced by r + cprp, in other words, r 1t, may be assumed. If A is defined as the direct sum of r rational groups Gi such that exactly t, of them satisfy p G , # G i and r of them satisfy p G i = G i , then it results easily that Char A is as desired.0

nNo

Another consequence which is sufficiently important to deserve a separate statement is due to S. Kakutani.

Corollary 47.5. The character group of a group of infinite cardinality n is of power 2". It is straightforward to prove that n is just the sum of all cardinals mo(p), * * €(p), taken for every p , and thus the direct product of the groups (1) must be of cardinality 2 " . 0 a ,

Since the cardinality of the set of non-isomorphic groups of cardinality

6 n is at most 2" (n 2 No), it is clear from (47.5) that the set of nonisomor-

phic compact (abelian) groups of cardinality 5 2" is at most of cardinality 2". Moreover, we have the rather surprising fact [which also shows that-though compact topology restricts the group structure considerably-the group structure has practically no effect at all on the compact topologies on the group] :

Theorem 47.6 (Fuchs [S]). For every infinite cardinal n, there exist 2" nonisomorphic compact groups of power 2" which are algebraically all isomorphic. In the proof, we need a result from Chapter XII. Namely, t.here exist 2" pairwise nonisomorphic p-groups of cardinality n, moreover, they can be chosen so as to have isomorphic basic subgroups @ =; 0, Z(p") and the same final rank n. By (47.1), their character groups are isomorphic to m

By the Pontryagin duality theory, they are not isomorphic as topological groups [cf. next section1.O

VIII.

202

HOMOMORPHISM GROUPS

The last theorem should be compared with the other extreme case: the groupJ," admits a single compact topology [which is the finiteindex topology]. In fact, (47.1) shows that the only discrete group whose character group is algebraically isomorphic to J," is the group @,Z(p"). [We have assumed the generalized continuum hypothesis to conclude that 111 is uniquely determined.] The methods of this section enable us to describe the groups of homomorphisms into algebraically compact groups. In this connection the basic result is: Theorem 41.1 (Fuchs [9]). If C is algebraically compact, then for every group A , Hom(A, C) is algebraically compact. From (38.1) we know that C is a direct summand of a direct product of cocyclic groups. Hence Hom(A, C) is a direct summand of a group of type Hom(A, Ci)with cocyclic groups C i[cf. (43.2)]. If C i is finite cyclic, say of order p", then Hom(A, Ci)is p"-bounded, and hence algebraically compact. If C iis quasicyclic, then by (47.1), Hom(A, Ci)is likewise algebraically compact. Consequently, Hom(A, Ci)is algebraically compact, and hence the result.0

n

n

As we know, the algebraically compact groups can be characterized by complete systems of cardinal invariants, so by (47.7), the same can be hoped for Hom(A, C) for algebraically compact C. As a matter of fact, the invariants of Hom(A, C) can be computed by means of the invariants of C and certain ones of A . For details, we refer to the Exercises.

EXERCISES (a) Let C be an algebraically compact group and G a pure subgroup of A . Verify the isomorphism Hom(A, C) z Hom(G, C) 0Hom(A/C, C). (b) Char A E Char G 0Char A / G . 2. Let A , B be reduced p-groups. Give a necessary and sufficient condition for the (algebraic) isomorphism Char A r Char B. 3. (a) The additive group of reals admits infinitely many compact topologies under which it is a topological group. [Hint: Char(@. Q).] (b) For which A is Char A torsion-free and divisible? Verify the statement in (a) for such a Char A . 4. If A is the quotient group of ZKomod the direct sum @ Z , then 1.

A

= Char

@ (Q 0 Q / Z ) . KO

[Hint: 42, Ex. 7.1 5 . (Hulanicki [ 11) A nonzero divisible group

D

=

0 Q 0 0 0 Z(p") m

P

m,

48.

DISCRETE TORSION AND

0-DIMENSIONAL COMPACT GROUPS

203

admits a compact topology exactly if the following conditions are satisfied : (i) in is of the form 2" with infinite n ; (ii) inp is finite or of the form 2 " p , np infinite; (iii) in >= mp for every p . 6. If C is a complete group, then Hom(A, C) is the inverse limit of bounded groups. 7. If C is p-adic algebraically compact, and if B is a p-basic subgroup of A , then Hom(B, C) z Hom(A, C). 8. (Pierce [l]) If A z @ , Z ( p " ) and C E @ J ( p " ) , then Hom(A, C ) E p-adic completion of the direct sum of d(in, 11) copies of .Ip.[Hint: apply (44.4) to 0 A [ p ] A ?+ A + 0 and C.] 9. Determine the invariants of the algebraically compact group Hom(B, C), if B is a direct sum of cyclic p-groups and C = 0, Z(p"). Z(p"). Determine the structure of 10. Let A be torsion-free and C = 0, Hom(A, C ) . [Hint: take p-basic i n A.] 11. By making use of Exs. 7-10, determine the invariants of Hom(A, C) for algebraically compact C. 12. (Pierce [l]) Find the invariants of Hom(A, C) if A is an arbitrary p-group. [Hint: (46.4), Exs. 8 and 9.1 ---f

48.*

--f

DUALITY BETWEEN DISCRETE TORSION AND 0-DIMENSIONAL COMPACT GROUPS

In this section we continue the study of character groups of abelian groups, but we no longer disregard the topology. As a matter of fact, the topology carried by character groups is very essential in the surprising duality between discrete and compact abelian groups. This duality theory for the general case of locally compact abelian groups is due to L. S. Pontryagin and E. R. van Kampen, and is based on a deep theorem which guarantees the existence of sufficiently many characters for a compact group. A closer examination of the proof of the duality shows, however, that this result is not needed if we restrict ourselves to 0-dimensional compact groups and their duals; in fact, in this case rather routine topological arguments suffice to establish duality. Our aim is to prove duality in this special case, i.e., the duality between discrete torsion groups on one hand and totally disconnected [that is, 0-dimensional] compact groups on the other hand. All topological groups in this section are assumed to be Hausdorff. To avoid tedious repetition of notation, we agree in denoting by K the real numbers mod 1, i.e., the circle group equipped with the usual topology, and by A* the group of all continuous homomorphisms of the topological

204

VIII.

HOXOMORPHISM GROUPS

group A into K, i.e., the character group of A. A* is awarded the compact-open topology, i.e., a fundamental system of neighborhoods about 0 is formed by all sets of the form U(C, E) = {x E A* I xC c K,} where K, is the &-neighborhood of 0 in K , and C is a compact subset of A . Moreover, we shall suppose that E is so small that K, contains no subgroup # O of K. We start with a few lemmas which we shall neither state nor prove in full generality, only in the case where we need them. In the proofs, standard results on topological groups will be taken for granted. (a) If A is discrete torsion, then A* is a 0-dimensional compact group. The functionsf on A to K form a group K A ;this is compact in the product topology, as K is compact. For fixed a, b E A , define

H(a, b ) = { fK~AI f (a + 6 ) = f ( a ) + f ( b ) ) which is a closed subset of K A [being defined in terms of an equation]. Clearly, A* is the intersection of all H(a, b) if we let a, b run over all elements of A ; hence A* is closed in K Aand thus compact in the induced topology. The proof of compactness will be completed when we have shown that this topology is the same as the compact-open topology on A*. Since A is discrete, compactness means finiteness : C = { a l , . . . , a,). If V(C,e) denotes the neighborhood of 0 in KAwhere the coordinates in the components corresponding to the elements of C belong to K,, then A* n V(C, E ) = U(C, E). Since the V(C,e) form a fundamental system of neighborhoods of 0 in KA, the compactness follows. In order to establish 0-dimensionality, notice that no generality is lost if C is assumed to be a subgroup of A , A being torsion. If E is chosen as agreed upon, then the condition xC c K, amounts to XC = 0, i.e., U(C, e) = Ann C, the annihilator of C, defined as Ann C = {

x A*~ IxC = O } .

Thus A* has a fundamental system of neighborhoods about 0 consisting of subgroups U(C, e); these are then open and closed, and hence A* is 0-dimensional. (b) Zf A is discrete torsion and a # 0 in A, then there exists a character x of A such that Xa # 0 . If c E K is of the same order as a, then a tt c extends to a homomorphism ( a ) + K. The latter group is divisible, hence this homomorphism can be extended to a x: A -+K. (c) If A is discrete torsion and C is afinite subgroup of A, then C* E A*/Ann C.

48.

DISCRETE TORSION AND

0-DIMENSIONAL COMPACT GROUPS

205

Because of the divisibility of K , every character of C can be extended to a character of A , that is, the map A* 4 C* induced by the injection C -+ A is epic. Ann C consists of all x E A* that induce the 0 character on C, i.e., Ann C is the kernel of A* + C*. (d) If G is a 0-dimensional compact group, then its topology is linear. Let U be an open-closed neighborhood of 0. For every U E U there is a neighborhood W , of 0 such that u + W , c U , and there is one V , such that V , V , c W,. Clearly, U is covered by (u + V,), hence by the compactness of U , there is a finite set {ul, ..., u,} such that (ui+ V,,) contains U . If V = n i V , , , then

u,,

+

u + I/ c IJ i

(Ui

UyIl

+ V,,, + V ) G u (Ui + W,,) E u . i

Let W be a neighborhood of 0 satisfying W = - W C U n V . Then also W + W G U + V G U , and by a simple induction, W + W + - . - + W S U. This shows that ( W ) s U. Since ( W ) is the union of the open sets W + ... + W , it is an open subgroup, and the assertion follows. (e) If G is a group with a linear topology, then a homomorphism y : G K is continuous if and only if Ker y is open. If Ker y is open, then G/Ker y is discrete and so y is obviously continuous. Conversely, if y is continuous, then y-’K, is open in G, whence there is an open subgroup H of G such that Ker y 2 H . Hence Ker y is open. --f

(f) If G is a 0-dimensional compact group, then G* is a discrete torsion group. If y E G*, then from (d) and (e) we see that Ker y is open, and therefore G/Ker y is discrete. It is compact, too, hence finite. Thus Im y is a finite subgroup of K, and so ny = 0 for some integer n > 0. This proves G* torsion. Now U(G, E ) = 0; thus G* is discrete. (g) If G is a 0-dimensional compact group and g # 0 in G, then there is a y E G* satisfying yg # 0. By (d), there exists an open subgroup H of G excluding 9. Now G/H is a discrete compact group, hence finite. (b) completes the proof. The assertions of (b) and (g) express the fact: “the groups A and G have sufficiently many characters.” This is fundamental in the duality.

The second character group A** = (A*)* of a group A contains characters a** of A* induced by elements a E A in the following fashion: a**(X) = Xa

for X E A * .

In view of the definition of addition of characters, a** is in fact a character of A* and the canonical map $: a w a * * of A into A** is plainly a homomorphism.

206

VIII.

HOMOMORPHISM GROUPS

(h) If A is a discrete torsion group, then the canonical map 4 : A isomorphism.

-+

A** is an

First we prove 4 monic. If a** = 0, then xa = 0 for all x E A*. In view of (b), this can happen only if a = 0 ; thus 4 is monic. If A is finite, then it is a direct sum of finite cyclic groups, and since Z(n)* Z(n), we have A E A*, and a repeated application gives A z A**. Thus 4 is an isomorphism for finite A. If A is any discrete torsion group, then, for a character y : A* -+ K , Ker y must be open because of (a), (d) and (e). Consequently, there is a finite subgroup C of A such that Ann C 5 Ker y [see the proof of (a)]. Now y induces a character 7 of A*/Ann C 2 C* [cf. (c)], and C* being finite, we know from what has already been proved that is induced by some element c E C, i.e., Y ( j ) = j c for all j E C*. It is now readily checked that y(x) = xc for all x E A* which proves y is of the form y = c**. Thus 4 is epic, and hence an isomorphism. (j) I f G is a 0-dimensional compact group, then the canonical map $: G -+ G** is a topological isomorphism.

That $ is monic follows in the same way as in (h), except that the reference to (b) must be replaced by that to (g). To prove ) I continuous, we show that if U is a fundamental neighborhood of G**, then $ maps some open subgroup Vof G into U . Now G* is discrete torsion by (f), so by the proof of (a), we may write U = Ann C for some finite subgroup C = {yl, , y,} of G*. Put V = Ker y j . By (d) and (e), Ker y j is open, and so is V . Now for g E Vwe find g**(yj) = y j g .= 0 for j = 1, * , n, that is, g** E Ann C. This proves that II,maps Vinto U , i.e., $ is continuous. Hence $ is a topological isomorphism between the compact groups G and $G. Since $G is compact, it is closed in the group G**. Now G**/$G is again 0-dimensional compact. If it is not 0, then application of (g) guarantees the existence of a nonzero character ?: G**/$G -+ K . This is induced by a character T : G** -+ K such that r$G = 0. By the discrete case (h), there is a x : G* -+ K such that z = x**, in other words, ~(y)= y(x) for all y E G**. Choose y = g** (with g E G) to obtain xg = g**(x) = z(g**) = 0. It follows that x = 0 and z = 0, in contradiction to Z # 0. Therefore $G = G**. To sum up, we have proved: Theorem 48.1 (Pontryagin [l]). Let A be a discrete torsion [0-dimensional compact] group. Then the group A* of its continuous characters is a O-dimensional compact [discrete torsion] group, and the correspondence a Ha** of A into the second character group A** is a topological isomorphism.m

207

NOTES

EXERCISES 1.

2.

3. 4.

5.

6. 7.

8. 9.

(48.1) continues to hold if characters are taken into the discrete group Q/z. Let A i (i E Z) be discrete torsion groups, and let @ A i have the discrete topology. Then ( @ A i ) * is topologically isomorphic to n A T equipped with the product topology. If G is a compact topological group and H is a closed subgroup, then (G/H)* z Ann H . Let G be a 0-dimensional compact group. (a) The Z-adic topology of G is finer than the given topology. (b) For every n E Z, n > 0, the subgroups nG and G[n] are closed. A group A is compact in its Z-adic topology if and only if it is complete in the Z-adic topology and A/pA is finite for every prime p . If a group is linearly compact i n its Z-adic topology, then it is compact in this topology. [Hint: Ex. 5.1 A group A is locally compact in the Z-adic topology if and only if for some n > 0, nA is compact i n the Z-adic topology. If A is discrete or compact, then Ann nA z A * [ n ] and Ann A [ n ] 2 nA*. For any compact topological group C, Hom(A, C) can be made into a compact topological group. NOTES It has long been known that the homomorphisms of a n abelian group into another form

a group. The importance of Horn was recognized by Eilenberg and Mac Lane [I], and Hom

as a fundamental functor was fully developed by Cartan and Eilenberg [I]. The idea of stating in (44.4) more than exactness for (2) and (3) whenever more than mere exactness is assumed for ( l ) , seems to appear first in the author’s papers, Fuchs [8] and [9]. Various other generalizations of (44.4) and (51.3) were given by Harrison et a/. [I], Irwin e / 01. [l], Pierce [l], and others. The algebraic structure of Hom was known in some special cases. A major contribution was made by Pierce [l], who computed the invariants of Honi(A, C ) as an algebraically compact group for torsion groups A . The same for Char A was done earlier by the author (Fuchs [S]), slightly improving the algebraic description of compact groups due to Hulanicki [l], [2] and Harrison [l]. [An excellent presentation of compact and locally compact abelian groups may be found in E. Hewitt and K. A. Ross, “Abstract Harmonic Analysis,” Vol. I, where the reader will also find more about dualities.] A remarkable duality between discrete and linearly compact p-adic modules has been discovered by I. Kaplansky [Proc. Amer. Mafh. SOC.4 (1953), 213-2191 and H . Schoneborn [Mrrtli. Z . 59 (1954), 455-473, and 60 (1954), 17-30]. It turns out that every linearly compact abelian group is i n a natural way a module over the Z-adic completion of the ring of integers [this ring is the direct product of the rings ofp-adic integers, one ring for every prime p]. The structure of linearly compact abelian groups can be completely described; they form a class between algebraically compact and compact groups (see Fuchs [13]).

WII.

208

HOMOMORPHISM GROUPS

Problem 30. Describe Hom(A, C), in particular, if C is torsion-free [of rank 1 or a direct sum of groups of rank 11. Torsion-free groups of rank 1 are characterized in Chapter XIII.

Problem 31. Find conditions on a group to be of the form End A for some A . How many of these A may be nonisomorphic? Notice that for torsion A , (46.4) leads to a solution.

Problem 32. Does Horn@, C) have a torsion groups A ?

"

natural " compact topology for

Problem 33. Which classes of abelian groups [subclasses of algebraically compact or cotorsion groups] A are closed under the correspondences A H Hom(G, A ) where G can be arbitrary? Problem 34. Does there exist a set X of groups X such that Hom(A, X ) z Hom(B, X ) for every X E X implies A z B? Problem 35. For which categories (d, I) and (d, D) are Hom(A, C I I) and Hom(A I D, C) always algebraically compact whenever A is torsion? Problem 36. Investigate the sets of endomorphisms of Hom(A, C) induced by those of A and C, respectively [common elements, centralizers, etc.]. Problem 37. Do there exist, for every infinite cardinal m,2" nonisomorphic compact and connected groups of cardinality 5 2 " [that are algebraically isomorphicl?

IX GROUPS OF EXTENSIONS

The extension problem for abelian groups [as a special case of the general group theoretical problem formulated by 0. Schreier] consists in determining the group from a subgroup and the corresponding quotient group. The classical way of discussing extensions is via factor sets. It was a profound discovery of R. Baer's, that the extensions-under a suitable equivalence relation-themselves formed a group Ext, the group of extensions. It is the study of this group which is our main topic in this chapter. An intimate relationship between groups of extensions and groups of homomorphisms has been pointed out by Eilenberg and Mac Lane [l]; this led to the interpretation of Ext as the socalled derived functor of Horn, and has been exploited extensively by Cartan and Eilenberg [I]. Ext can be discussed in various ways; we shall rely upon the elegant method of Mac Lane [3]. Our main objective is to discover the group theoretical properties of Ext (C, &its dependence upon the groups A and C , and its relation with known constructions. Some of our goals are beyond the limits of our methods, but a good deal of information can be obtained about the general case which settles the problem in a number of special cases. The exact sequences connecting Horn and Ext, their generalizations, and the theory of cotorsion groups are the most relevant results in this chapter.

49. GROUP EXTENSIONS

Given the groups A and C, the extension problem consists in finding groups B such that B contains a subgroup A' isomorphic to A and B/A' r C. This situation can be expressed in terms of a short exact sequence E :0

--f

A

B L. C+ +0

where p stands for the inclusion map and v is an epimorphism with kernel P A . In this case, one says that B is an extension of A by C, and it is our present

209

Ix.

210

GROUPS OF EXTENSIONS

aim to survey all extensions of A by C. This can be done in various ways. In this section we describe the extensions in terms of factor sets, while in the next section we present the discussion based on short exact sequences. Let a, b, . . . denote elements of A , and u, u, w , . . * those of C. Let g : C -+ B be a representative function, i.e., g(u) is a representative of the coset u, g(u) E v-lu. Every b E B can be written uniquely in the form b = g(u) pa with a E A . Clearly, g(u) + g(u) and g(u + u) belong to the same coset mod P A , hence there is anf(zc, u) E A such that

+

(1)

g(4

+ g(4 = s(u + 4 + P f ( % 4.

Thus we have a function

(2)

f:CxC+A

which is uniquely determined by the extension B and the choice of the representatives g(u). The commutative and associative laws in B imply the identities (3) (4)

fb, u) =f(u, 4, S(u, 0) + f (u + u, 4 =f (4u + 4 +f ( u , w)

for all u, u, w E C. Choosing g(0) = 0, we have in addition f ( U , 0 ) =f (0,

(5)

2’)

=0

for all u, u E C. A function ( 2 ) satisfying (3)-(5) is said to be a factor set [on C to A ] . Assume, conversely, we are given two groups, A and C, together with a factor set (2). We can construct a group B as the set of all pairs (u, a) E C x A with the operation (u, a)

+ ( u , b) = (u + u, a + b + f ( u ,

0)).

In fact, the commutative and associative laws are consequences of (3) and

(4),while (0,O) is the zero and ( - u , -a -f ( - u , u)) is the inverse to (u, a) in B. Manifestly, 1-1 : a H (0, a) and v : (u, a) Hu make the sequence 0 + A”+

c-+

BL+ 0

exact, thus B is an extension of A by C. The choice g(u) = (u, 0) yields the given factor set f. Thus the extension problem can be soloed by deterniining all factor sets. The direct sum C @A is one of the extensions of A by C ; it is also referred to as the splitting extension. If g(u) = u E Cis chosen, then f is identically 0. Another choice, say, g(u) = u + ph(u) with h(u) E A yields the factor set (6)

f ( u , V)

= h(u)

+ h(u) - h(u +

0).

49.

21 1

GROUP EXTENSIONS

Conversely, if h : C -+ A is any function with h(0) = 0, then (6) is a factor set, such that the pairs (u, -h(u)) form a complement to pA in B, i.e., the extension splits. Thus an extension of A by C is splitting exactly if it is defined in terms of a factor set f of the form (6) [where-we emphasize-h : C -t A ] ; such an f is called a transforniation set. The dependence of the extension upon the chosen function g : C B can be excluded by introducing the following equivalence relation in the set of extensions. If gl, g, are both representative functions C -t B, then clearly g,(u) - g,(u) = ph(u) for some function h : C A . The corresponding factor setsf,, f, satisfy --f

--f

(7)

fl(u,v )

-f2(11,

L)) = h(u)

+ h(U) - h(u + .).

Accordingly, we define the factor sets f,, f, : C x C + A equitlalent, if (7) holds for some h : C A . The extensions B,, B, of A by C corresponding to equivalent f,,f, are then isomorphic under the correspondence b : (u, a)l w (u, a + h(u)), which makes the diagram -+

E , :O-tA-!!-+B,'+C-+O

commutative. I n this case, the extensions El and E, themselves are called equitlalent. Considering that if (8) is commutative, and if g1 : C-t B, is a representative function, then g 2 = Dgl : C B, is also one, and the corresponding factor sets are equivalent, we conclude that there is a one-to-one correspondence betH'een the equiidence classes of extensions of A by C and the equicalence classes of factor sets f:C x C -+ A . In particular, an extension is equivalent to the splitting extension if, and only if, the corresponding equivalence class of factor sets is the class of transformation sets. -+

Iff,, f, : C x C -+ A are factor sets, then their sum fi+ fi defined as

(f,+f,)(k 0 ) =f1(u, 4 +fz(u,V) is again a factor set, and so is -,fi too. Consequently, the factor sets on C to A form a group Fact(C, A ) . The transformation sets form obviously a subgroup Trans(C, A ) of this group, and what has been shown above can also be formulated by asserting a one-to-one correspondence between the equivalence classes of extensions of A by C and the elements of the quotient group Fact(C, A)/Trans(C, A ) . This quotient group is the group of extensions of A by C : Ext(C, A ) = Fact(C, A)/Trans(C, A ) . Another method of discusssion i s based on producing a presentation of the extension in terms of A and a presentation of C . This method will be outlined briefly at the end of 51.

Ix.

212

GROUPS OF EXTENSIONS

EXERCISES 1. If Hom(C, A ) is regarded as a subgroup of A'' [cf. 431, then A''/Hom(C, A ) E Trans(C, A )

where

C'

=

C\O.

2. A factor set on any group to a divisible group is a transformation set. [Hint : (22. l).] 3. Every factor set on C to A is equivalent to a factor set on C to a subgroup B of A , if AIB is divisible. 4. If A E C E Z(p), then there are two nonisomorphic and (at least) p nonequivalent extensions of A by C. 50. EXTENSIONS AS SHORT EXACT SEQUENCES

In the preceding section we used the method of factor sets to describe the extensions of a group A by another one C. Another approach is based upon short exact sequences. This decisive new idea will lead to a number of instructive relations, as we shall now see. If the extension B of A by C is visualized as an exact sequence O-+A"-

B A C + O ,

then one can try to build up a category in which the objects are just the short exact sequences. An adequate definition of a morphism between two exact sequences is rather clear: it is a triple (a, p, y) of group homomorphisms such that the diagram E : O - + A "B'-C+O

has commutative squares. It is straightforward to show that in this way a category d arises. In accordance with the previous definition of equivalent extensions, we say that the extensions E and E' with A = A', C = C' are equivalent, in sign: E = E', if there is a morphism ( l a , p, 1,) with p : B -+ B' an isomorphism. Actually, the condition of p being an isomorphism can be omitted, since this follows already from (2.3). First we study extensions with A fixed. If y : C' -+ Cis any homomorphism, then to the extension E in (l), there is, by (lO.l), a pullback square B'

C'

PI

1 y

O+A&B--%C-+O

50.

EXTENSIONS AS SHORT EXACT SEQUENCES

213

with suitable B',B and v'. From 10 (a) we know that v' is epic [since v is epic], and a glance at (3) in 10 shows that Ker v' r Ker u E A , hence there is a monomorphism 11' : A + B' [namely, p'u = (pa, 0) E B' if B' 6 B 0C'] such that the diagram

with exact rows and pullback right square commutes. The top row is an extension of A by C' which we have denoted by Ey to indicate its origin from E and y. Notice that y* = (l,, fi, y ) is a morphism Ey --* E in 6. If the diagram

E": O+A

Bo?+

I

E: O+A " . B L + C

C' + 0 1 y

+O

has exact rows and commutes, then by (10.1) there is a unique 4 : B" -+ B' such that v'4 = v o and fl4 = Do. Since the maps 4 p o , p' : A -+ B' are such that P(4p") = pOpo = p = fip' and v'(4p") = vopo = 0 = v'p', the uniqueness assertion in (10.1) implies 4 p o = p'. Hence (I,, 4, lc.) is a morphism of E" to Ey, and so E" E Ey. This shows that Ey is unique up to equivalence, and this yields the equivalences El,

=E

and

E(yy') = (Ey)y'

for C" L+ C' >+ C. Now the contravariance of E on C is evident. Next we keep C fixed and let A vary. Given a : A + A', let B' be defined by the pushout square O-+AABY-C-+O

Here p' is a monomorphism, since p is one [cf. 10(b)]. Moreover, if B' is defined as a quotient group of A' 0 B [as in the proof of (10.2)], then v'((a',6 ) + H = vb makes the diagram

214

Ix.

GROUPS OF EXTENSIONS

with exact rows commutative. The bottom row is an extension of A' by C which we have denoted by aE. Here a* = (a, p, 1,) is a morphism E -+ U E in 8. If E : O+A-'-+B'C+O

is a commutative diagram with exact rows, then in view of (10.2) there exists a unique 4 : B' + B, such that 4p = Po and 4p' = po . From (v, 4)p = v, p, = v = v'p, (v, 4)p' = 0 = v'p' we infer that v, 4 = v', thus ( I A , , 4, 1,) is a morphism ME+ E, . Consequently, uE = E, , i.e., aE is unique up to equivalence. Hence 1, E E E and (acl')E EE a(a'E) for A L + A ' - + A " , establishing the covariant dependence of E on A . With a : A -+ A' and y : C' -+ C we have the important associative law (2)

cr(Ey) s (aE)y.

Indeed, by making use of the pullback property of (aE)y, it is easy to prove the existence of a morphism (a, p', 1) : Ey --f (aE)y and to show the commutativity of the square Ev

Let us pause for a moment to point out that both Ey and aE can be described easily in terms of factor sets, if E is given by a factor setf: C x C -+ A . Owing to the definition of Ey in terms of pullback, the factor set belonging to Ey is the composite function C' x C r z +C x C-+f

A,

as is readily seen from the fact that B @ C' is an extension of A by C 0 C' with the factor set f(cl + c;, c2 + c;) =f(cl, c2) (ciE C, cf E C') and restriction to B' means c1 = yc;, c2 = yc; , i.e., the factor set isf(yc;, yci). Similarly, the factor set describing aE is the composite function C x C - +f A A + A ' . Indeed, in aE, B' is a quotient group of the extension A' @ B of A' @ A by C with the factor setf(c,, c 2 )viewed as C x C -+ A' @ A ;passing to the quotient

50.

215

EXTENSIONS AS SHORT EXACT SEQUENCES

group, the pushout property shows that B‘-as an extension of A’ by Cwill have the factor set aJ(cl, c,). Returning to short exact sequences, assume we are given two extensions El and E , of A by C. As has been shown in the preceding section, the extensions of A by C [more correctly, their equivalence classes] form a group. In order to describe the group operation in the language of short exact sequences, we make use of the diagonal map AG : g H ( g , g) and the codiagonal map V , : (gl, g 2 ) w g l g 2 of a group G. If we understand by the direct sum of two extensions

+

E i: 0 + A++

Bi-%

c., + o

( i = 1,2)

the extension

El @ E 2 : 0 4 A , @A,-+

B, @ B2-

VI@V2

C, @ C, + 0,

we then have: Proposition 50.1 (Mac Lane [3]). The sum of two extensions E l , E, of A by C is the extension (3)

El

+ E , = VA(E1 0 E,) A c .

What we have t o verify is that iff, : C x C + A is a factor set belonging to Ei(i = 1, 2), then fl+f, belongs to V,(E, 0 E,) Ac . Clearly, (fl(cl, c,), f,(c;, c;)) with c i, c; E C is a factor set belonging to the direct sum El 0E, , and (fl(cl, c,), f2(c1, c,)) is one corresponding to (El 0E 2 ) A c . An application of V , yields the factor set fl(cl, c 2 ) + f,(cl, c 2 ) . o It is of course possible to avoid any reference to factor sets and to develop extensions solely qua short exact sequences. In doing so, (3) would serve as the definition of the sum of extensions and then (50.1) should be replaced by the assertion that E , E2 is actually an extension of A by C which stays in the same equivalence class if E , and Ez are replaced by equivalent extensions, and moreover, the equivalence classes of extensions form a group under this operation. [For a proof of this, without using factor sets, we refer to Mac Lane

+

PI.] From what has been said above about the factor sets belonging to Ey and ciE it is now evident that for homomorphisms ci : A -+ A’ and y : C‘ 4 C, the following equivalences hold true for extensions E l , E, , E of A by C: (4) (5)

+ EZ) EEL + M E , , (al + a,)E = a l E + a,E,

“(El

+ E,)y E1y + E,y, E(y, + y2) = Ey, + E y 2 .

(El

The equivalences (4) express the fact that a* : E H rE and y* : E H Ey are group homomorphisms a* : Ext(C, A ) -+ Ext(C, A ’ ) ,

y* : Ext(C, A ) + Ext(C‘, A ) ,

Ix.

216 while (5) assert that (a1 the correspondence

GROUPS OF EXTENSIONS

+ a2)* = (a1)* + (a2)* and (yl + y2)* = yT f yf , i.e.,

Ext : C x A H Ext(C, A ) , y x a t+ y*a*

= a,y*

is an additive bifunctor on d x d to d [the last equality is just another form of (2)]:

Theorem 50.2 (Eilenberg and Mac Lane [l]). Ext is an additive bijiinctor on d x d to d which is contravariant in thefirst and covariant in the second variab1e.n In order to be consistent with the functorial notation for homomorphisms, we shall also use the notation Ext(y, a): Ext(C, A) -+ Ext(C’, A‘) instead of y*a*

= a,y*;

that is, Ext(y, a) acts as shown by Ext(y, a ) : E w a E y .

Let us keep in mind that if the extension E is given by (I), then for -,C , Ey is represented by 0 -+ A?‘+ B ‘ L C’ -+ 0 where

y : C‘

(6)

B‘

=

{(b, c’) I b E B, c’ E C’, vb

and for a : A

--t

A’,

p’a = (pa, 0),

aE is represented by 0 -+ A‘”+B‘&C-+

B‘

(7)

= yc’},

=

{(a’, 6 ) + HI a’

p ’ ~= ’ (u’, 0 )

+ H,

E

v’(b, c’) = c’,

0 where

A’, b E B } ,

~ ‘ ( ( ab’ ), + H ) = vb

with H = { ( N U , -pa) I a E A } . These formulas for Ey and aE are helpful in subsequent computations. EXERCISES Characterize the mono- and epimorphisms of the category & [i.e., the morphisms that are left- and right-cancellable]. 2. (Mac Lane [3]) If E is as in ( l ) , then both p E and Ev split. 3. (Mac Lane [3]) If (a,b, y ) : E + E’ is a morphism in the category 8, then aE = E’y. 4. (a) If a : A A’ is an epimorphism, then aE [with E in (l)] is equivalent to the extension 1.

--f

0

--f

with the obvious maps.

A/Ker a

-+

B / p Ker a -+ C-+O

5 1.

EXACT SEQUENCES FOR EXT

217

(b) If y : C' + C is a monomorphism, then Ey is equivalent to O-+ A

-+

v-'

Im y

-+

Im y +O.

5. Let ct [ y ] be an automorphism of A [C]. When is ctE [Ey] equivalent to E in ( l ) ? 51. EXACT SEQUENCES FOR EXT

As we have seen in the preceding section, Ext is a functor in both of its variables. The main result of this section states that this functor is right exact, moreover, the exact sequences on Horn and Ext can be amalgamated into long exact sequences. Given a n extension

E : o -+ A &

(1)

BP+ c -+ 0,

representing a n element of Ext(C, A ) , and a homomorphism q : A -+ G, we know from the preceding section that qE is a n extension of G by C , i.e., qE represents an element of Ext(C, G). In this way we get a map E* : Hom(A, G) + Ext(C, G) defined as E* : q w q E .

Analogously, a homomorphism A by G, and

5 : G + C yields

from E an extension E( of

E , : Hom(G, C) + Ext(G, A ) is a m a p acting as follows: E, :