Infinite Abelian groups.

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The theory had been first developed in the paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra. The fundamental theorem of finite abelian groups states that every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special case of the fundamental theorem of finitely generated abelian groups when G has zero rank. The cyclic group Z mn of order mn is isomorphic to the direct sum of Z m and Z n if and only if m and n are coprime.

Infinite Abelian Groups

It follows that any finite abelian group G is isomorphic to a direct sum of the form. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic. One can apply the fundamental theorem to count and sometimes determine the automorphisms of a given finite abelian group G. Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p -subgroups separately that is, all direct sums of cyclic subgroups, each with order a power of p.

Fix a prime p and suppose the exponents e i of the cyclic factors of the Sylow p -subgroup are arranged in increasing order:. In this case the theory of automorphisms of a finite cyclic group can be used. Here, one is considering P to be of the form. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so. In the most general case, where the e i and n are arbitrary, the automorphism group is more difficult to determine.

It is known, however, that if one defines. One can check that this yields the orders in the previous examples as special cases see [Hillar,Rhea]. The simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary orders. Even though the decomposition is not unique, the number r , called the rank of A , and the prime powers giving the orders of finite cyclic summands are uniquely determined.

By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible Baer's criterion. An abelian group without non-zero divisible subgroups is called reduced. An abelian group is called periodic or torsion if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known.

These theorems were later subsumed in the Kulikov criterion. An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:. An abelian group that is neither periodic nor torsion-free is called mixed. Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. One of the most basic invariants of an infinite abelian group A is its rank: the cardinality of the maximal linearly independent subset of A.

Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of Q and can be completely described. More generally, a torsion-free abelian group of finite rank r is a subgroup of Q r. On the other hand, the group of p -adic integers Z p is a torsion-free abelian group of infinite Z -rank and the groups Z p n with different n are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.

The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups.

Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings with nontrivial multiplication. Some important topics in this area of study are:. The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab , the prototype of an abelian category.

Nearly all well-known algebraic structures other than Boolean algebras are undecidable. Bounded Pure Subgroups Quotient Groups Modulo Pure Subgroups Pure-Exact Sequences Pure-Projectivity and Pure-Injectivity Generalizations of Purity Notes VI.

Basic Subgroups Algebraically Compact Groups Algebraic Compactness Complete Groups Pure-Essential Extensions Homomorphism Groups Groups of Extensions Group Extensions Extensions as Short Exact Sequences 5 I. Exact Sequences for Ext Elementary Properties of Ext The Functor Pext Cotorsion Groups The Structure of Cotorsion Groups The Ulm Factors of Cotorsion Groups Applications to Ext Tensor and Torsion Products Some of the proofs will be omitted as they are standard and can be found in textbooks on algebra or on group theory.

Structure theorems for infinite abelian groups

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. 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. Let P be a partially ordered set, i. I f a partially ordered set is inductive,then it contains a maxip a l element.

On autocommutators and the autocommutator subgroup in infinite abelian groups : Forum Mathematicum

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. 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. 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.

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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. 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. This may be finite or infinite; in the first case, B is offinite index in A. The elements of A contained in elements [i. 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. 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 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. 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.

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Clearly, nA and A[n] are subgroups of A. The zero is of 1. 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. 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. The set of all subgroups of a group A is partially ordered under the inclusion relation.

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Prove that a finite group A contains an element of order p if and only if p divides the order of A. Let B, C be subgroups of A. If A contains elements of infinite order, then the set of all elements of infinite order in A generates A. Bn C and Find examples where proper inclusions hold. A is called the domain and B the range or 2.

A map a : A preserves addition, that is,. One may write aA for Im a. As customary in algebra, we make no distinction between isomorphic groups, unless they are distinct subgroups of the same larger group considered.

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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.

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