A group whose operation is commutative (cf.
Commutativity).
They are named after
N.H. Abel,
who used such groups in the theory
of solving algebraic equations by means of radicals. It is customary
to write the operation in an Abelian group in additive notation,
i.e. to use the plus sign
(
)
for that operation, called
"addition" ,
and the zero sign
for the neutral element,
named zero (in multiplicative notation it is called the unit element).
Examples of Abelian groups.
All cyclic groups (cf.
Cyclic group)
are Abelian, in particular, the additive group of integers. All direct sums (cf.
Direct sum)
of cyclic groups are Abelian. Also the additive group of rational numbers
is Abelian; it is moreover a
locally cyclic group,
i.e. a group in which all finitely generated
subgroups are cyclic. Finally, the groups of type
(or the quasi-cyclic groups
),
where
is an arbitrary prime number, are Abelian (cf.
Group of type
;
Quasi-cyclic group).
The
free composition
in the variety of Abelian groups coincides with the direct sum. A
free Abelian group
is a direct sum of infinite cyclic groups. Every
subgroup of a free Abelian group is free Abelian. The set
of all elements of finite order in an Abelian group forms a subgroup, which is called the
torsion subgroup
(periodic part)
of the Abelian group.
The quotient group of an Abelian group by its torsion subgroup is a
group without torsion.
Thus, every Abelian group is an extension of a torsion-free
group by a torsion group. This extension does not always
split, i.e. the torsion group is usually not a direct summand.
An Abelian torsion group in which the order of every
element is a power of a fixed prime number
is said to be
primary
with respect to
(the term
-group
is used in general group theory). Every torsion
group splits uniquely into a direct sum of
primary groups that correspond to distinct prime numbers.
A complete classification is known for finitely
generated Abelian groups. This is given by the
fundamental theorem of finitely generated Abelian groups:
Every finitely generated Abelian group is a direct sum of finitely many
non-split cyclic subgroups some of which are finite and primary, while
the others are infinite
(G. Frobenius,
L. Stickelberger).
In particular,
finite Abelian groups split into a direct sum of primary
cyclic groups. Such splittings are, in general, not unique, but
any two splittings of a finitely generated Abelian
group into direct sums of non-split cyclic groups are
isomorphic, so that the number of infinite cyclic summands and
the collection of the orders of the primary cyclic summands
do not depend on the splittings chosen. These numbers are called
invariants of the finitely generated Abelian group.
They constitute a complete system of invariants, in the sense
that two (finitely generated) Abelian groups are isomorphic if and only
if they have the same invariants. Each subgroup of
a finitely generated Abelian group is itself finitely generated.
Not every Abelian group is a direct sum of
(even infinitely many) cyclic groups. For primary groups there
is a necessary and sufficient condition for the
existence of such splittings, the so-called Kulikov criterion. Let
be a primary Abelian group for a prime number
.
A non-zero element
of
is said to be an
element of infinite height
in
if for any integer
the equation
is solvable in
,
and
is called an
element of height
if this equation is solvable for
only. The
Kulikov criterion
states that a primary Abelian group
is a direct sum of cyclic groups if and only
if it is the union of an increasing sequence of subgroups such that
the set of heights of the elements in each one of
these subgroups is bounded. Any subgroup of an Abelian group that
is a direct sum of cyclic groups itself is
such a direct sum. The indecomposable (non-splittable) into
direct sums primary Abelian groups are the
primary cyclic groups
and the group
.
A finite set of elements
in an Abelian group is called
linearly dependent
if there exist integers
,
not all equal to zero, such that
.
If such numbers do not exist, the set is said to be
linearly independent.
An arbitrary collection of elements of
is said to be linearly dependent if there exists a finite
linearly dependent subcollection. An Abelian group that is not
a torsion group has maximal linearly independent sets. The
cardinality of all maximal linearly independent collections of
elements is the same and is called the
rank
(Prüfer rank)
of the given Abelian group. The rank of a torsion group is considered to
be zero. The rank of a free Abelian group coincides with
the cardinality of a set of free generators of it.
Every torsion-free Abelian group of rank 1 is isomorphic to some subgroup
of the additive group of rational numbers. There exists a complete
description of such groups in the language of types. Each
element of an Abelian group without torsion determines a
characteristic,
which is a countable sequence consisting of non-negative numbers and the symbol
.
This sequence is constructed in the following manner. Suppose
that the prime numbers are enumerated in increasing order
.
An element
determines the sequence whose
-th
element is equal to
if the equation
is solvable in the group but the equation
is not solvable, and the
-th
element is equal to
if
is solvable for all
.
Two characteristics are equivalent if they are equal except possibly
for a finite number of places and if the symbol
occurs in exactly the same places in both sequences. The characteristics of
two linearly dependent elements are equal. A class
of equivalent characteristics is said to be a
type.
To each torsion-free Abelian group of rank 1
there corresponds a uniquely determined type, called the
type of the given group;
non-isomorphic groups having different types.
A torsion-free Abelian group that splits into a direct sum
of groups of rank 1 is said to be
completely split.
Not every subgroup of a completely split group is
completely split, though every direct summand is. For each integer
there exists a torsion-free Abelian group of rank
that cannot be decomposed (or split) into a
direct sum. For countable torsion-free Abelian groups one
can to construct a complete system of invariants.
An Abelian group is called
complete,
or
divisible,
if for any one of its elements
and for any integer
the equation
has a solution in the group. All divisible Abelian groups
turn out to be direct sums of groups isomorphic to
and the groups
,
and the cardinalities of the sets of components isomorphic to
,
as well as to
(for each
),
from a complete and independent system of invariants of
the divisible group. Every Abelian groups can be isomorphically
imbedded in some divisible Abelian group. The divisible Abelian groups and
only they are the injective objects in the category of Abelian
groups. A divisible Abelian group is a direct summand of each
Abelian group containing it. Therefore, an Abelian group is
a direct sum of a divisible Abelian group and a so-called
reduced group,
i.e. a group that contains no non-trivial
divisible subgroups. A classification of reduced Abelian groups
is known only in certain special cases. Thus,
Ulm's theorem
([1])
gives the classification of all countable reduced Abelian torsion groups.
The theory of Abelian groups has its origins in
number theory, and is now extensively applied in many
modern mathematical theories. Thus, the duality theory of characters
for finite Abelian groups has been considerably extended to
the duality theory of locally compact Abelian groups. The development of
homological algebra has made it possible to solve a whole
series of problems in Abelian groups, such as classifying the
set of all extensions of one group by another. The
theory of modules is closely connected with Abelian groups regarded
as modules over the ring of integers. Many results in the theory
of Abelian groups can be applied to the case of modules
over a principal ideal ring. Owing to their relative simplicity and
to the fact that they have been very thoroughly studied
(which is confirmed, for instance, by the solvability of the
elementary theory
of Abelian groups), and to the availability of a sufficient
variety of objects, Abelian groups serve as a constant
source of examples in various fields of mathematics.