I Preliminaries

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

5

1. DEFINITIONS

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 and

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

(Bn C)

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

I.

8

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

has

--f

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

lYI 1.; 1.. 1.

A , >-+

A2&

A,&

A,&

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.

11

MAPS AND DIAGRAMS

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

I.

12

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. If 0 + A L + B B + C -+ 0 is an exact sequence, and B‘ 5 B, then there 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.

15

THE MOST IMPORTANT TYPES OF GROUPS

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,.

16

I.

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)

<

. + *

< (c,) < . . . )

since to a given n there exists one and only one subgroup of order p", namely, that generated by c, .

Theorem 3.1. A group C is cocyclic if and only if C g Z ( p k ) with 1, 2, - - .or 00. Let c E C denote a cogenerator of C. Then (c) is the smallest subgroup fO of C , and therefore c has to be of prime order p. Since c lies in every nonzero subgroup of C, C contains neither elements of infinite order nor elements whose order is divisible by a prime f p , i.e., C is a p-group. As a basis of induction, assume that C contains at most one subgroup C, of order p" and this is cyclic, C, = (c,). If A , B are subgroups of C of order p n f l ,and if a E A\C, , b E B\C, ,then a, b 4 C,implieso(a) = p n C 1= o(b). Thuspa = rc, . pb = sc, for suitable integers r , s prime top. If r ' , s' are such that rr' = 1 = ss' mod p", then Y', s' are prime to p , and a' = r'a, b' = s'b satisfy: ( a ' ) = ( a ) , pa' = c,, ( b ' ) = ( b ) , pb' = c, . Hence p(a' - 6') = 0, a' - b' = tc,, for some integer t , that is, a' = b' + tpb', 6' = a' - tpa', and a', b' generate the same cyclic group ( a ' ) = ( b ' ) . Consequently, A = ( a ) = ( b ) = B, and so C is the union of a finite or infinite ascending chain of subgroups of orders p", i.e., C is of the form Z(pk).O k

=

Evidently, all the quasicyclic groups belonging to the same prime p are isomorphic. Since the subgroups of Z(p") are of type Z(pk),the quotient groups f O of Z(p") are seen to be again Z(p"). The group of all complex roots of unity, i.e., the group of all rotations of finite order of the circle, obviously contains Z(p") for every prime p as a subgroup. It has the remarkable property of being locally cyclic in the sense that all of its finitely generated subgroups are cyclic. It contains all finite cyclic groups as subgroups.

Rational groups. Under addition the rational numbers form a group called thefull rationalgroup, denoted by Q. Like Z(p"), Q can also be obtained as a

3.

17

THE MOST IMPORTANT TYPES OF GROUPS

union of an infinite ascending chain of cyclic subgroups: z=( 1 ) < ( 2 ! r 1 ) < ... < (n!-') < ....

Thus Q has a generating system c,, . . . , c, , . . . satisfying

(2)

2 C 2 = c1,

3C, = c 2 , . ' . , (n

+ l)c,+, = C " , ....

It is easy to see that Q is locally cyclic, too: every finite set of elements is contained in some (n!-'); therefore, the subgroup they generate is a subgroup of a cyclic group, and so is itself cyclic. Q contains numerous proper subgroups which are not finitely generated, as the group Q, of all rational numbers with denominators prime to p , or the group Q(,) of all rational numbers whose denominators are powers of p . The subgroups of Q , called rational groups, are of fundamental importance in the theory of torsion-free groups. Every proper quotient group Q / A of Q (i.e., A # 0) is readily seen to be a torsion group, since every rational number has a multiple in A . In particular, Q / Z is isomorphic to the group C of all complex roots of unity, an isomorphism being induced by the epimorphism r w e 2 I r n[where r E Q , i = and e is the base of natural logarithms] of Q onto C, whose kernel is 2. More generally, Q / ( r ) E Q / Z for every rational r # 0, while, e.g., Q / Q , E Z ( p " ) .

fl,

p-adic integers. The p-adic integers have many applications in various branches of abelian group theory. Let us sketch a method of introducing the p-adic integers. Let p be a prime and Q, the ring of rational numbers whose denominators are prime to p . The nonzero ideals of Q, are principal ideals generated by p' with k = 0, I , . . . [i.e., it is a discrete valuation ring]. If the ideals ( p h )are considered as a fundamental system of neighborhoods of 0, then Q, becomes a topological ring, and we may form the completion Q f of Q , in this topology [this completion process is described in detail for groups in 131. Q,* is again a ring whose ideals are ( p k )with k = 0, I , . . . , and which is complete [i.e., every Cauchy sequence in Q f is convergent] in the topology defined by its ideals. The elements of Q f may be represented as follows: let { t o , t , , ..., be a complete set of representatives of Q, mod pQ,, e.g., (0, 1, . . . ,p - 1); then {p't, , p't,, . . . ,p k t , - is one of pkQp mod pk+'Q,. Let n E Q f , and let u,EQ,, be a sequence tending to n. Owing t o the definition of Cauchy sequences, almost all a,, [i.e.. all with a finite number of exceptions] belong to the same coset mod pQ,, e.g., to the one represented by s o . Almmt all differences a, - so belonging to p Q , belong to the same coset of p Q , mod p 2 Q p , say, to that represented by p s , . So proceeding, n uniquely :'dines ii sequence so, s,p, s2p2,... , and we assign to n the formal infinite seri-\ so + s , p + s , p 2 .... Its partial sums b,=s,+s,p+ ...+s,,p" (n=l,2,-..)

+

I.

18

PRELIMINARIES

form a Cauchy sequence in Q, which converges in Q,* to n, in view of n - b, €pkQp*(for n 2 k ) . From the uniqueness of limits, it follows that, in this way, different elements of Q; are associated with different series, and since every series so + slp + s2p2+ * * . with coefficients in a fixed system of representatives defines a n element of QZ, we may identify the elements n of Q: with the formal series so slp s2p2 . . ., with coefficients from {ro, r,, . - . ,r p - l } , preferably from {0, 1, " . , p - l}, and write

+

(3)

n =so

+

+ s,p + ... + s,p" + ...

+

(with s,

= 0,

1,

. . . ,p

- 1).

The arising ring Qp* is a commutative domain [where domain is a ring without divisors of zero] called the ring of p-adic integers; its cardinality is the power of the continuum. Notice that if p = ro + r,p + ... + rnpn+ ... (r, = 0, 1, " * , p - 1) is another p-adic integer, then the sum n p = qo + q,p . .. q,p" . . . and the product np = qh q ; p + . .. + qApn + ... are as follows: qo = so ro -k,,p,q~=sOrO-mOp,q~=sOr,,+s,r,~, -kop,q,=s,+r,+k,-l s,,ro m,- I - m, p (n = I , 2, .. .), where the integers k , , k , , m, , m, are uniquely determined by the fact that all of q, and 4;are between 0 and p - 1. As to subtraction and division, note that the negative of n = s, p" + ~ , + ~ p " + ~ + ... (s, # 0) is - n = ( p - s,)pn (p - s , + ~ - l ) p n + l + . . . ,and the inverse n-l of (3) exists if and only if so # 0; it may be found by using the inverse rule to multiplication. For the additive group of Q; we shall use the symbol J , .

+

+

+

+

+

+

+

+ +-.-

+

EXERCISES 1. A simple (abelian) group is isomorphic to Z(p) for some prime p. 2. A has a composition series if and only if A is finite. 3. (a) A subgroup M of A is called maximal if M < A and M 5 B < A implies M = B. Show that M is maximal if and only if it is of prime index. (b) Prove that Z(p"), Q have no maximal subgroups; Z(pk)(k = 1,2, . . -), J, have exactly one maximal subgroup; and Z has infinitely many maximal subgroups. 4. (a) The intersection of all maximal subgroups of A of the same prime index p is PA. (b) The Frattini subgroup of A [i.e., the intersection of all maximal subgroups of A ] is the intersection of allpA withp running over all primesp. (c) What are the Frattini subgroups of Z(n), Z , Z(p"), Q, Q p ,Jp? 5. Prove that neither Z(p") nor Q can be finitely generated. 6. (a) Show that the group of all complex roots of unity is locally cyclic.

4.

19

MODULES

(b) Every subgroup and every quotient group of a locally cyclic group is again locally cyclic. 7. Prove the isomorphisms:

Q / Q , g Z(p">, for k = 1, 2, .

P'"'/Z z Z(p">,

Jp/pkJp Z(pk)

8. (L. Rtdei) If a group A contains subgroups isomorphic to any one of Z(pk)with a fixed prime p and k = 1,2, * , but n o proper subgroup of A has this property, then A r Z(p"). [Hint: p A = A and select generators.] 9. I n a cyclic group A , two subgroups B and C coincide if A / B E A/C. 10. (a) Prove that a p-adic integer n: of the form (3) is a p-adic unit if and only if so # 0. (b) The field of quotients of Q: consists of all elements of the form zp-" with n: E Q*, and n a nonnegative integer. 1 .

4.

MODULES

Most of the theorems in abelian group theory can be generalized mutatis rriutandis to unital modules over a principal ideal domain R with identity, and everything can be carried over-without any modification in the proofs -if R has the additional property that all quotient rings R/(a) with 0 # a E R are finite. It is, however, a delicate question to find the natural boundaries of a particular theorem in abelian groups, i.e., to describe the class of rings such that, for the modules over these rings, the theorem in question holds. A discussion of problems like this is beyond o u r present subject, and therefore we shall restrict ourselves to abelian groups only, i.e., modules over the ring Z of integers. Occasionally, however, we have to consider modules, since they yield a natural method of discussion. Therefore, let us recall the definition of modules. Let R be an associative ring and M a n abelian group such that (i) with CY E R and a E M there is associated an element of M , called the product of ci and a, and denoted by ma; (ii) (./?)a = .(/?a) for all a, /3 E R and a E M ; (iii) a(a b ) = aa +ab for all a E R, a, b E M ; (iv) (a /?)a= cia +/?a for all ci, /? E R, a E M .

+ +

I n this case, M is said to be a left R-module or a left module over R. If R has a unit element E , then it is in most cases assumed that E acts as the identity operator on M : (v)

EU =

a for all a E M .

I.

20

PRELIMINARIES

Such R-modules M a r e called unital. In our discussions, we shall only consider unital R-modules where R will always be commutative, in which case there is no need to distinguish between left and right R-modules. [In the Notes, modules are unital left modules.] Recall that a submodule N of a n R-module M is defined to be a subset of M which is an R-module under the same operations, i.e., it is a subgroup of M such that aN L N for all a E R. In this case, the quotient group M / N becomes a n R-module, the quofienl module, where a(a + N ) = aa + N for all cosets a + N and a E R. If M , N are R-modules, then a group homomorphism 4 : M -+ N is said to be an R-homomorphism if it satisfies for all a E M ,

4(aa) = a4(a)

c1 E

P.

The meaning of R-isomorphism, etc., is obvious. For a in a n R-module M , the order o(a) is defined as the set of all annihilators of a in R: ~ ( a= ) {a E

R I uu = O } .

Thus o(a) is a left ideal of R. The case o(a) = (0) corresponds to elements of infinite order in groups. Example 1. If R is the ring Z of integers, then every abelian group A can be regarded as a Z-module under the natural definition of multiplication of n t Z and a E A , namely, nu is the nth multiple of a . Example 2. If R is the ring Q, of rationals with denominators prime top, then a p-group

A can be made into a Q,-module in a natural way. Namely, if ( n . p ) = 1, then for every

a E A, the product n - l a is a uniquely determined element of .Indeed, if r , s are integers such that nr o(a)s = 1, then a = nra o(a)sa = nra shows that n - l a = ra [and it is easy to see that in A no element f ra gives a on multiplication by n].

+

+

Example 3. In a similar way, we conclude that every p-group A is a Q:-module in . E Q:, and if a E A is of order p", then the natural way: if n = so i-s l p . . . s,p"

+ +

Ta = (so

+

+ slp + . . . + s,-lpn-l)a,

where the element o n the right does not change if we use a larger partial sum of T . Let us notice that in all of our examples, submodules, R-homomorphisms are simply subgroups, group homomorphisms.

Modules over

QE

are also called p-adic modules.

EXERCISES 1. The cyclic R-module generated by a is R-isomorphic to the R-module R/o(a).

5.

CATEGORIES OF ABELIAN GROUPS

21

2. If R is a principal ideal domain, if M is an R-module, and a, b E M such that o(a) = (a) # 0 and o(b) = (j?) # 0, then o(a + b) is a divisor of aj?(cc, j?)-’ and a multiple of @(a, fl)-2. 3. Let R be a commutative domain and M an R-module. Then the elements a E M such that o(a) # 0 form a submodule N of M such that in the quotient module M / N all the elements # O have order 0. 4. Let N be a submodule of an R-module M . Prove that o(a) s o(a + N ) where o(a + N ) denotes the order of the coset a + N in M / N . 5. If R = Z/(m),m an integer >0, then for every element a in an R-module we have ma = 0. 6 . If $I : R’ + R is a ring-homomorphism carrying the identity of R’ into that of R, then every R-module A4 becomes an R’-module by putting a’a = $I(cc’)a for all a’ E R’, a E M . 7. A Q-module is-as a group-torsion-free. 5. CATEGORIES OF ABELIAN GROUPS

In the theory of abelian groups, it is often convenient to express situations in terms of categories. In fact, categories and functors seem to be proper unifying concepts in a number of cases. Therefore, let us introduce categories and exhibit some important concepts connected with them. Categories are not algebraic systems in the usual sense of the word, i.e., they are not necessarily sets equipped with algebraic operations. Categories need not even be sets, they aremerely ‘‘classes.’’ It isnecessary to get rid of the assumption of being sets, since,e.g., we often consider all abelian groups which do not form a set. [Therefore, it is apt to use the Godel-Bernays axioms of set theory, where both sets and classes are admitted. However, if one wishes to avoid the use of classes, then he may restrict himself to abelian groups belonging to some “universe” in the sense of Grothendieck.]

A category %? is a class of objects A , B, C , .. ., and morphisms a, j?, y, . . . satisfying the following axioms: I . With each ordered pair A, B of objects in %? there is associated a set Map(A, B) of morphisms in %? such that every morphism in %? belongs to exactly one Map ( A , B). If a E %‘ belongs to Map(A, B) then we write a : A + B and may call a a map of A into B, while A is the domain, B the range of a . 2. With a E Map(A, B) and j? E Map(B, C ) , there is associated a unique element of Map(A, C ) , called their product Pa. 3. Whenever the products are defined, associativity prevails: 4. For each A E 8 ‘ there exists a morphism 1, E Map(A, A ) , called the identity morphism of A, such that 1, a = a and PI, = fl whenever the products make sense.

I.

22

PRELIMINARIES

One verifies at once that 1, is uniquely determined by the object A ; indeed, if ,z E Map(A, A ) has the same property, then z,1, must be equal both to 1, and to .1, Calling a morphism i E %' an identity if za = a and = p whenever the products are defined, we conclude that there is a one-toone correspondence between the objects A and the identities 1, of %?, and, therefore, categories can also be defined in terms of morphisms only. There are numerous examples for categories: the sets with mappings as morphisms, the not necessarily commutative groups or rings with homomorphisms as morphisms, R-modules with R-homomorphisms, topological spaces with continuous mappings, etc. For our present topic, the most important example is the category d of all abelian groups where the objects are the abelian groups, and the morphisms are the homomorphisms between them. Obviously, the torsion (abelian) groups, the p-groups, the torsion-free groups, et al. form categories if the morphisms are again the homomorphisms. [In general, if the objects of some category are groups, then it is implicitly understood, unless otherwise stated, that the morphisms of this category are just the homomorphisms between the groups in the category.] Just as we have homomorphisms between algebraic systems, correspondingly we have functors between categories. If %? and 9 are categories, then a covariant functor F:%?-+9

az

[on %? to 91 assigns to each object A E %3an object F(A) E 9, and to each morphism a : A + B in %? a morphism F(a) : F(A)-+ F ( B ) in 9 satisfying the following conditions : (i) if the product Pa of a, p E %' is defined, then [F(p)F(a) is defined in 9 and] F(Po0 = F ( P ) W ;

(ii) F carries the identity of A E V into that of F ( A ) E 9,i.e., for all A€%?, F(1,) = I F ( , ) .

Thus a covariant functor preserves domains, ranges, products, and identities. The identity functor E, defined by E(A) = A , E(a) = a for all A , ci E %',is a covariant functor on V to itself. A contravariant functor G : %? 9 is defined similarly by reversing arrows, i.e., G assigns an object G(A) E 9 to every object A E 2' 7, and a morphism G(a) : G(B) -+ G(A) in 9 to every morphism a : A + B in 59,and it is subject t o the conditions -+

G(Pa) = G(a)G(B),

The unqualified term

"

= IC(A).

functor " will usually mean covariant functor.

5.

23

CATEGORIES OF ABELIAN GROUPS

If F is a functor on %? to 9, and G is a functor on 9 to a category 6 , then the composite GF is a functor on 59 to 6' [where G F ( A ) = G(F(A)), and GF(a) = G(F(c()),for all A , CY E %?I. Clearly, GF is covariant if F , G are both co- or both contravariant, and is contravariant if one of F , G is co-, while the other is contravariant. We shall have to consider functors in several variables, covariant in some of their variables and contravariant in others. For instance, if 59, 9, € are categories, then a bifunctor F on x 9 to 6, covariant in %? and contravariant in 9,assigns t o each couple (C, D ) with C E %, D E 9 an object F(C, D ) E & a n d to each pair CY : A + C, J : B -+ D of morphisms (CY E 59, J E 9) a morphism F(cq p) : F ( A , 0 )+ F(C, B ) such that (1)

F(yCY,sp> = 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

*

Im $

with t+b : X

--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.

P<"

[Evidently, for every CL : A -+ B, F"(cL)is the restriction of c1 to F"(A).]If, for instance, F is the functor V, in (a), then F" is just U as in (b), where o stands for the first limit ordinal. The Ulm subgroup U ( A )of A , introduced in (b), will be of great importance later on. An alternative definition is U(A) =

nn ~ . n

The equivalence with the definition given in (b) follows from results in Chapter

111. More precisely, U ( A ) is called t h e j r s t Ulrn subgroup of A , while the oth

iterated functor U"(A) = A" yields the 0th Ulrn subgroup of A . The quotient group U"(A)/U"+'(A)= U"(A) = A ,

(o = 0, 1,2, . . .)

is said to be the 0th Ulrn factor of A ; in particular, A/A' = A , is the 0th Ulrn factor of A . For a discussion of Ulrn subgroups and factors, we refer to 37. Notice that if F , , F , are subfunctors of the identity, and F , 5 F , , then the inclusion maps 6" : F,(A) F,(A) yield a natural transformation cD : F , -+ F , , and the corresponding maps 42 : A / F , ( A ) --* A / F , ( A ) define a natural transformation cD* : FT -+ FT. EXERCISES 1. Prove that for subfunctors F , G , H of the identity, we have (FG)H = F(GH), FG 2 F A G, and for F 5 H , F v (G A H ) = ( F v G ) A H . 2. If T, S are as in examples 1 and 2, then TZ= T, S2 = S , TS = ST = S.

7.

29

TOPOLOGIES IN GROUPS

3. For a subfunctor F of the identity, C 5 A implies F(C) 5 C n F ( A ) 4.

(F(A)

+ C)/C 5 F ( A / C ) .

Let X and YJ be two classes of groups. (a) If X E 9,then V, 5 V , and W, 5 W,. (b) One has always V,,,,

5.

and

=

V,

A

and

V,

WE,,,, = W, v W,, .

For a class X of groups, let X, and X, denote the class of all subgroups and quotient groups, respectively, of groups in X. Then VES =

V,,

v,,

5 V,

3

Wx, = W,,

WXS

2

K'

6 . If F is a subfunctor of the identity, and p , Dare ordinals, then FPF" = F u f p . 7. For every class X and for every pair p , CJ of ordinals, with p 5 a, v x A / V ; ; ( A ) )= V $ ( A ) /V ; ; ( A ) and C ( A / W ; ; ( A ) )= W$(4/W;;(A)

hold for every A E d. 8. Let F , be subfunctors of the identity where CJ ranges over all ordinals less than an ordinal z. Define the infinite product .. . F , . . . F I F O(a < z), and show that this is likewise a subfunctor of the identity. 9.* By making use of the notion of direct sums [see 81 verify the formula

for a subfunctor F of the identity. Show that the analog fails to hold for direct products. 7. TOPOLOGIES IN GROUPS

In abelian groups, topology can be introduced in various ways which are natural in one sense or another. These topologies are not necessarily Hausdorff; some of them do not even make the group operations continuous. The importance of topologies in abelian groups will be evident from subsequent developments. The most important topologies to be considered here are linear topologies; that is to say, there is a base [fundamental system] of neighborhoods about 0 which consists of subgroups such that the cosets of these subgroups form a base of open sets. In order to formulate our definition in a reasonably general

I.

30

PRELIMINARIES

fashion, let us start with a dual ideal (i.e.,filter) D in the lattice L(A) of all subgroups of A, and declare the subgroups U of A in D as a base of open neighborhoods about 0. Then the cosets a U ( U E D) will be a base of open sets about a. Because the intersection of two cosets (al + U,) n (a, + U,) is vacuous or a coset mod U , n U,, and U , n U , E D whenever U , , U, E D, all the open sets will be unions of cosets a + U with U E D. The continuity of the group operations is obvious from the simple observation that x - y E a U implies (x U ) - ( y + U ) c a U. We call the arising topology the D-topology of A. Obviously, it is Hausdorfs if and only if

+

+

+

+

n u=o,

UED

and discrete exactly if 0 E D. Occasionally, one has a topology which satisfies the first axiom of countability, i.e., there is a countable base of open neighborhoods about 0. If U,, U 2 , . .. , U,, , * . * is such a system of neighborhoods, then U,, U , n U, , . - - ,U, n n U,, . * * is also one; thus, in this case we may without loss of generality assume the sequence decreasing. An open subgroup B is necessarily closed. Indeed, its complement in the group is the union of its cosets a + B (a $ B ) ; these are open and so is their union. Hence the subgroups U E D are both open and closed. Therefore: 1 . .

Proposition 7.1. I f D is a dual ideal in the lattice L(A) of all subgroups of A , and f t h e D-topology of A is Hausdorfs, then it makes A into a 0-dimensional topological group. 0 The following special cases are of significance. 1. The 2-adic topology where the subgroups nA (n E Z , n # 0) form a base of neighborhoods about 0. This is a D-topology where D consists of all U S A such that A / U is bounded. The 2-adic topology is Hausdorff exactly if n n A = 0, n

i.e., the first Ulm subgroup U ( A ) of A vanishes. This topology is discrete if and only if nA = 0 for some n, i.e., A is bounded. It is easy to see that the closure B - of a subgroup B is given by the formula B-

=

n(B+ nA), n

and B is a closed subgroup exactly if the first Ulm subgroup of A/B vanishes. 2. The p-adic topology arises if only the subgroups p k A (k = 0, 1,2, . .) are declared to belong to the base of neighborhoods at 0. [Now the dual ideal D is the set of all U 5 A such that A / U are bounded p-groups.] We have the following simple result.

7.

31

TOPOLOGIES IN GROUPS

Theorem 7.2. The following conditions on a group A are equivalent: (a) the p-adic topology of A is Hausdorff, (b) A contains no elements # 0 of injnite p-height ; (c) llall = exp( - h,(a)) is a norm on A (a E A ) ; (d) 6(a, b) = Ila - bll is a metric on A (a, b E A ) that yields the p-adic topology. Since the p-adic topology of A is Hausdorff if and only if r ) k p k A = 0 , the equivalence of (a) and (b) is evident. (b) implies that h,(a) = cxis equivalent to a = 0, while referring also to hp(a + b) 2 min(h,(a), h,(b)), we have at once: (i) llall 2 0 for every a E A ; (ii) llalj = 0 is equivalent to a = 0; (iii) Ila bjl 5 max(llall, 11b11) for all a, b E A . Thus llall is actually a norm on A . In order to see that (c) implies (d), notice that the norm properties imply that 6(a, b) = Ila - bll is a metric on A , in which a base for open sets consists of spheres of radius E ( E >O), 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 ) .

I.

32

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

+

+

m

rn

i=l

i= 1

+

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

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

+ S,)

n

i= 1

(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.].