Page 120 - DMTH403_ABSTRACT_ALGEBRA

Unit 10: Finite Abelian Groups

Associatively Notes

For all a, b and c in A, the equation (a • b) • c = a • (b • c) holds.

Identity Element

There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds.

Inverse Element

For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity
element.
Commutatively

For all a, b in A, a • b = b • a.
More compactly, a finite abelian group is a commutative group. A group in which the group
operation is not commutative is called a “non-finite abelian group” or “non-commutative group”.

You should notice that any field is a finite abelian group under addition. Furthermore, under
multiplication, the set of non-zero elements of any field must also form a finite abelian group.
Of course, in this case the two operations are not independent–they are connected by the
distributive laws.
The definition of a finite abelian group is also useful in discussing vector spaces and modules.
In fact, we can define a vector space to be a finite abelian group together with a scalar multiplication
satisfying the relevant axioms. Using this definition of a vector space as a model, we can state the
definition of a module in the following way.

10.2 Properties

Let us assume that, If n is a natural number and x is an element of a finite abelian group G written
additively, then nx can be defined as x + x + ... + x (n summands) and (–n)x = –(nx). In this way, G
becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with
the finite abelian groups.
Theorems about finite abelian groups can often be generalized to theorems about modules over
an arbitrary principal ideal domain. A typical example is the classification of finitely generated
finite abelian groups which is a specialization of the structure theorem for finitely generated
modules over a principal ideal domain. In the case of finitely generated finite abelian groups,
this theorem guarantees that a finite abelian group splits as a direct sum of a torsion group and
a free finite abelian group. The former may be written as a direct sum of finitely many groups
of the form Z/p Z for p prime, and the latter is a direct sum of finitely many copies of Z.
k
If f, g : G  H are two group homomorphisms between finite abelian groups, then their sum
f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism. (This is not true if H is a non-
finite abelian group.) The set Hom (G, H) of all group homomorphisms from G to H thus turns
into a finite abelian group in its own right.
Somewhat kind to the dimension of vector spaces, every finite abelian group has a rank. It is
defined as the cardinality of the largest set of linearly independent elements of the group. The
integers and the rational numbers have rank one, as well as every subgroup of the rationals.

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