Starting with examples of abelian groups, the treatment explores torsion groups, Zorn's lemma, divisible groups, pure subgroups, groups of bounded order, and direct sums of cyclic groups. Subsequent chapters examine Ulm's theorem, modules and linear transformations, Banach spaces, valuation rings, torsion-free and complete modules, algebraic compactness, characteristic submodules, and the ring of endomorphisms. Many exercises appear throughout the book, along with a guide to the literature and a detailed bibliography.
Starting with examples of abelian groups, the treatment explores torsion groups, Zorn's lemma, divisible groups, pure subgroups, groups of bounded order, and direct sums of cyclic groups. Subsequent chapters examine Ulm's theorem, modules and linear transformations, Banach spaces, valuation rings, torsion-free and complete modules, algebraic compactness, characteristic submodules, and the ring of endomorphisms. Many exercises appear throughout the book, along with a guide to the literature and a detailed bibliography.
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Overview
Starting with examples of abelian groups, the treatment explores torsion groups, Zorn's lemma, divisible groups, pure subgroups, groups of bounded order, and direct sums of cyclic groups. Subsequent chapters examine Ulm's theorem, modules and linear transformations, Banach spaces, valuation rings, torsion-free and complete modules, algebraic compactness, characteristic submodules, and the ring of endomorphisms. Many exercises appear throughout the book, along with a guide to the literature and a detailed bibliography.
Product Details
| ISBN-13: | 9780486836454 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 12/19/2018 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 112 |
| File size: | 3 MB |
About the Author
Read an Excerpt
CHAPTER 1
1. INTRODUCTION
In the early days of group theory attention was confined almost entirely to finite groups. But recently, and above all in the last two decades, the infinite group has come into its own. The results obtained on infinite abelian groups have been particularly penetrating. This monograph has been written with two objectives in mind: first, to make the theory of infinite abelian groups available in a convenient form to the mathematical public; second, to help students acquire some of the techniques used in modern infinite algebra.
For this second purpose infinite abelian groups serve admirably.
No extensive background is required for their study, the rudiments of group theory being sufficient. There is a good variety in the transfinite tools employed, with Zorn's lemma being applied in several different ways. The traditional style of transfinite induction is not completely ignored either, for there is a theorem whose very formulation uses transfinite ordinals. The peculiar role sometimes played by a countability hypothesis makes a challenging appearance.
It is furthermore helpful that finite abelian groups are completely known. In other subjects, such as rings or nonabelian groups, there are distracting difficulties which occur even in the finite case. Here, however, our attention is concentrated on the problems arising from the fact that the groups may be infinite.
With a student audience in mind, I have given details and included remarks that would ordinarily be suppressed in print. However, as the discussion proceeds it becomes somewhat more concise. A serious effort has been made to furnish, in brief space, a reasonably complete account of the subject. In order to do this, I have relegated many results of some interest to the role of exercises, and a large part of the literature is merely surveyed in the guide to it provided in § 20.
This material is adapted from a course which I gave at the University of Chicago in the fall of 1950. I should like to record my indebtedness to the many able members of that class, particularly to George Backus, A rlen Brown, and Roger Farrell. Thanks are expressed to Isidore Fleischer for the ideas in § 16 (the torsion-free case of Theorem 22 was discovered by him and appears in his doctoral dissertation); to Robert Heyneman and George Kolettis, who read a preliminary version of this work and made many valuable suggestions; to Tulane University and the University of Michigan, where I had the opportunity to lecture on abelian groups; and to the Office of Naval Research.
A special acknowledgment goes to Professor Reinhold Baer. It was from his papers that I learned much of the theory of abelian groups. Furthermore, when this monograph was nearly complete, I had the privilege of reading an unpublished manuscript (of book length) on abelian groups which he prepared in the late 1940's.
2. EXAMPLES OF ABELIAN GROUPS
Before beginning to develop the theory, it is desirable to have at hand a small collection of examples of abelian groups.
To avoid endless repetition, let it be agreed that "group" will always mean "abelian group."
(a) Cyclic groups. A group G is cyclic if it can be generated by a single element. If that element has infinite order, G is isomorphic to the additive group of integers, and is called an infinite cyclic group; if it has finite order n, G is cyclic of order n and is isomorphic to the additive group of integers mod n. We shall use the notation Z and Zn respectively for these two groups.
(b) External direct sums. Let {Gi} be any set of groups, where the subscript i runs over an index set I, which may be finite or infinite. We define the direct sum of the groups Gi. We take "vectors" {ai}; that is, arrays indexed by i [member of] I with ai in Gi. Moreover, we impose the restriction that all but a finite number of the ai's are to be 0 (we are writing 0 indifferently for the identity element of any Gi. Addition of vectors is defined by adding components. This gives an abelian group, called the direct sum of {Gi}.
If there is any danger of ambiguity, the object just defined may be referred to as the "weak," or "discrete," direct sum, as opposed to the "complete" direct sum, where the vectors are unrestricted. In pure· algebra it is the weak direct sum which arises most naturally; the complete direct sum is, indeed, mostly useful as a source of counterexamples (see Theorem 21 and exercise 33).
(c) Union and intersection. If S and T are subgroups of a group, we write S n T for their intersection, that is, the set of elements lying in both. More generally, if {Si} is a set of subgroups of G, we write [intersection] Si for the intersection. Note that we are talking about the set-theoretic intersection and that it is always a subgroup.
As regards the union of subgroups, the situation is different. Consider first two subgroups, S and T. The set-theoretic union, which we might write S [union] T, is not generally a subgroup (in fact, S [union] T is a subgroup if and only if one of the two subgroups S and T contains the other). What we wish instead is the smallest subgroup containing S and T, and this is provided by S + T, the set of all elements s + t, where s and t range over S and T.
Again, let {Si} be any set of subgroups of G. Their union, written ΣSi, is the smallest subgroup containing them; it may be explicitly described as the set of all finite sums of elements extracted from the various subgroups Si.
(d) Internal direct sums. In dealing with direct sums we are most often confronted with the problem of showing that a group is isomorphic to the direct sum of certain of its subgroups. Suppose first that the group G has subgroups S and T satisfying S [intersection] T = 0, S + T = G. Then it is easy to see that G is isomorphic to the direct sum of S and T, where we are referring to the external direct sum discussed above in (b). One may speak of G as being the internal direct sum of S and T, but generally one simply calls G the direct sum of S and T, and writes G = S [direct sum] T.
Consider now any (finite or infinite) set of subgroups (Si}. In order to verify that G is the direct sum of these subgroups, the most convenient procedure is generally as follows: Show that G = ΣSi, that is, that every element of G can be written as a finite sum of elements from the subgroups Si; then show that the representation is unique. This uniqueness is equivalent to the statement that each Si is disjoint from the union of the remaining ones.
In general, if the union ΣSi of subgroups is their direct sum, we shall call the subgroups Siindependent.
A concept of independence for elements will also be useful: We shall say that the elements xi are independent if the cyclic subgroups they generate are independent in the sense just defined, and we write Σ(xi) for the subgroup generated by all the elements.
We should notice the analogy between this concept and linear independence in a vector space. In fact, the elements xi are independent if and only if the following is true: If a finite sum
Σnixi = 0 (ni integers),
then each nixi = 0.
(e) Rational numbers. The most general group so far in our possession is a direct sum of cyclic groups. A classical theorem asserts that this covers all finitely generated groups, and in particular all finite groups. That is to say, any finitely generated group is a direct sum of (a finite number of) cyclic groups.
One might for a moment think that perhaps any abelian group is a direct sum of cyclic groups, the number of summands now being allowed to be infinite, of course. This conjecture is defeated by a very familiar group: the additive group R of rational numbers. That R is not a direct sum of cyclic groups may be seen, for example, from the fact that for any x [member of] R and any integer n there exists an element y E R with ny = x; this property manifestly cannot hold in a direct sum of cyclic groups. (The property in question is called divisibility, and will be studied in § 5.)
(f) Rationals mod one. In the additive group R of rational numbers, there is the subgroup Z of integers. The quotient group R/Z is known as the rationals mod one. We note that in R/Z every element has finite order. We argue, just as above, that R/Z is not a direct sum of cyclic groups.
(g) The group Z(p∞). There is an important modification of the two preceding examples. Let p be a fixed prime, and let P denote the additive group of those rational numbers whose denominators are powers of p. The quotient group P/Z will play a dominant role in the ensuing discussion, and we use for it the notation Z(p∞).
Let us pause to take a close look at Z(p∞). For simplicity we take p = 2. We can write the elements of Z(2∞) as 0, 1/ 2, 1/4, 3/4, 1/8, etc., but it is to be understood that addition takes place mod one. Thus
1/2 + 1/2 = 0, 1/2 + 3/4 = 1/4, 3/4 + 5/8 = 3/8, etc.
What are the subgroups of Z(2∞)? There is a subgroup of order 2 consisting of 0 and 1/2; one of order 4 consisting of 0, 1/4, 1/ 2, 3/4 ; and in general a cyclic subgroup (say Hn) of order 2n generated by 1/2n. It is not difficult to see that these are in fact the only subgroups. Thus the array of subgroups can be pictured as follows:
[MATHEMATICAL EXPRESSION OMITTED]
It is noteworthy that every subgroup of Z(2∞) is finite, except for Z(2∞) itself. The subgroups form an ascending chain which never terminates. On the contrary, one sees that every descending chain of subgroups must be finite. Thus Z(2∞) has the so-called "descending-chain condition" but not the "ascending-chain condition."
In conclusion, we give another realization of Z(p∞). Consider the set of all pn-th roots of unity, where p is a fixed prime and n = 0, 1, 2, .... These numbers form a group under multiplication, and the group is isomorphic to Z(p∞).
This completes our discussion of examples. It will appear that these groups are the fundamental building blocks for some fairly wide classes of infinite abelian groups.
3. TORSION GROUPS
If an abelian group has all its elements of finite order, we shall call it a torsion group. (This designation does not convey much algebraically, but it has a suggestive topological background and the merit of brevity.) The other extreme case is that where all the elements (except 0 of course) have infinite order; we then call the group torsion-free.
Now let G be an arbitrary abelian group, and T the set of all elements in G having finite order. We leave to the reader the verification of the following two remarks: (a) T is a subgroup, (b) G/T is torsion-free. We shall call T the torsion subgroup of G.
The study of abelian groups is now seen to split into three parts: (a) the classification of torsion groups, (b) the classification of torsion-free groups, (c) the study of the way the two are put together to form an arbitrary group. Progress has been most notable on the first of these problems, and consequently we shall be chiefly concerned with torsion groups.
Next we define a group (necessarily a torsion group) to be primary if, for a certain prime p, every element has order a power of p. The study of torsion groups is reduced to that of primary groups by the following theorem:
Theorem 1. Any torsion group is a direct sum of primary groups.
Proof. Let G be the group, and for every prime p define Gp to be the subset consisting of elements with order a power of p. It is clear that Gp is a subgroup, and that it is primary. We shall now prove that G is isomorphic to the direct sum of the subgroups Gp.
(a) We have first to show that G is the union of the subgroups Gp. Take any x in G, say of order n. Then factor n into prime powers: [MATHEMATICAL EXPRESSION OMITTED], and write [MATHEMATICAL EXPRESSION OMITTED]. Thus n1 ..., nk have greatest common divisor 1, and so there exist integers a1, ..., ak with a1n1 + ... + aknk = 1. Then
(1) x = a1n1x + ... + aknkx.
Now nix has precisely order [MATHEMATICAL EXPRESSION OMITTED], and so it is in [MATHEMATICAL EXPRESSION OMITTED]. Thus equation (1) is the desired expression of x as a sum of elements in the Gp's.
(b) We have further to prove the uniquen ess of the expression ju st found. Suppose
x = y1 + ... + yk = z1 + ... + zk
where yi, zi lie in the same [MATHEMATICAL EXPRESSION OMITTED]. Consider the equation
(2) y1 - z1 = (z2 + ... + zk) - (y2 + ... + yk).
We know that y1 z1 has order a power of p1. On the other hand, the right side of (2) is an element whose order is a product of powers of p2, ..., pk. This is possible only if y1 - z1 = 0. Similarly each yi = zi. This completes the proof of Theorem 1.
As a general principle, every decomposition theorem should be accompanied by a uniqueness investigation. Such an investigation is particularly easy for the decomposition given by Theorem 1. In fact, there is only one way to express a torsion group as a direct sum of primary subgroups, one for each prime p; for the subgroup attached to p must necessarily consist of all elements whose order is a power of p. In other words, the decomposition is unique not just up to isomorphism; the summands are unique subgroups.
The simplicity of the proof of Theorem 1 is a natural counterpart to this strong uniqueness; for if a decomposition is unique there ought to be a simple natural way to effect it. It is instructive to compare this situation with later ones. For example, under suitable hypotheses of various kinds we shall prove that a primary group is a direct sum of cyclic groups; this decomposition is unique, but only up to isomorphism. The difficulties encountered in the proof are a natural reflection of the large number of arbitrary choices that have to be made in carrying out the decomposition.
We shall conclude this section by giving two illustrations of Theorem 1:
(a) Consider the cyclic group Z(n), where [MATHEMATICAL EXPRESSION OMITTED]. Then [MATHEMATICAL EXPRESSION OMITTED]. (Indeed, this is the fact which really underlies the proof of Theorem 1.)
(b) Let G be the additive group of rationals mod one (§ 2). This is a torsion group, and it can be seen that its primary component for the prime p is precisely the group Z(p∞) of § 2. Thus G is a direct sum of all the groups Z(p∞).
4. ZORN'S LEMMA
Nearly every proof to follow will depend on the use of a transfinite induction. Such an induction is generally best accomplished by the use of Zorn's lemma, which is to be regarded as an axiom like other axioms needed to set up the foundations of mathematics.
We shall make use of a version of Zorn's lemma which refers to the concept of a partially ordered set. A partially ordered set is a set with a binary relation 2; which satisfies
(a) x [??] x (reflexivity),
(b) x [??] y, y [??] x imply x = y (antisymmetry),
(c) x [??] y, y [??] z imply x [??] z (transitivity).
Let S be a partially ordered set and T a subset. The element x is said to be the least upper bound of T if x [??] y for every y in T and if z [??] y for every y in T implies z [??] x. (The element x itself may or may not be in T.) A least upper bound need not exist, but if it does, it is unique.
An element x of a partially ordered set S is said to be maximal if S contains no larger element. It is to be observed that S may contain many maximal elements.
A partially ordered set is a chain (also called a simply ordered set or a linearly ordered set) if every two elements are comparable; that is, either x [??] y or y [??] x.
We now state Zorn's lemma:
Zorn's lemma. Let S be a partially ordered set in which every chain has a least upper bound. Then S has a maximal element.
This brief account will suffice for the applications we shall make of this lemma. We refer the reader to the literature for details on other forms of Zorn's lemma, and their equivalence to the well-ordering axiom or the axiom of choice.
5. DIVISIBLE GROUPS
In an abelian group any element may be multiplied by an integer. But what about dividing by an integer? The answer is that the result may not exist, and if it exists, it may not be unique. So we shall not attempt to attach a meaning to the symbol 1/nx, but nevertheless we shall say that x is divisible by n if there exists y with ny = x.
Examples. (a) The element 0 is divisible by any integer.
(b) If x has order m, then it is divisible by any integer prime to m.
(c) In the additive group of rational numbers, every element is divisible by every integer.
In this section we are going to study groups which share this last property with the additive group of rational numbers.
Definition. A group G is divisible if for every x in G and every integer n there exists an element y in G with ny = x.
Alternatively, G is divisible if G = nG for every integer n.
We note that a cyclic group is not divisible. Nor for that matter is a direct sum of cyclic groups. Indeed, it is clear that a direct sum of groups is divisible if and only if every summand is divisible. Another easily verified fact is that a homomorphic image of a divisible group is divisible. So we note that the group of rational s mod one is divisible, since it is a homomorphic image of the additive group of rationals.
(Continues…)
Excerpted from "Infinite Abelian Groups"
by .
Copyright © 1969 The University of Michigan Press.
Excerpted by permission of Dover Publications, Inc..
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Table of Contents
1. Introduction2. Examples of Abelian Groups
3. Torsion Groups
4. Zorn's Lemma
5. Divisible Groups
6. Two Test Problems
7. Pure Subgroups
8. Groups of Bounded Order
9. Height
10. Direct Sums of Cyclic Groups
11. Ulm's Theorem
12. Modules and Linear Transformations
13. Banach Spaces
14. Valuation Rings
15. Torsion-free Modules
16. Complete Modules
17. Algebraic Compactness
18. Characteristic Submodules
19. The Ring of Endomorphisms
20. Notes
Bibliography
Index