abelian group

In group theory, abelian is used to describe a group for which, in addition to the usual group axioms, the group operation * commutes- that is, the ordering of elements is unimportant (more formally, ∀a,b∈G, a*b = b*a). Sometimes referred to as commutative groups, the expression abelian is in honour of the Norwegian mathematician Niels Henrik Abel, yet despite this it is usual to write abelian rather than the grammatically correct Abelian (much to the consternation of one of my linear algebra lecturers).

Given any field F, F is abelian under addition and F^x= F\{0} is abelian under multiplication. An example of a non-abelian group is given by the general linear group GL(n,F), since matrix multiplication is not generally commutative.

A number of useful tests/properties hold for the abelian groups and related subgroups. For a group described in the form of a multiplication table, the abelian property will hold if the table is symmetrical about the diagonal, such as for Z₂ X Z₂:

   e   a   b   c
e| e   a   b   c
a| a   e   c   b
b| b   c   e   a
c| c   b   a   e

Why should this be the case? The entry in the i,j^th cell corresponds to the operation of * on the i^th and j^th group elements, i.e. g_i*g_j. But for an abelian group, g_i*g_j=g_j*g_i so the same value must appear in the j,i^th cell- hence the symmetry arises.

Testing is even easier in certain cases-any group of order 5 or less is abelian. Also, any cyclic group is necessarily abelian- for G=<g> consider that for any a,b∈G there are m,n such that a=g^m, b=gⁿ, then ab=g^mgⁿ=g^m+n=g^n+m=gⁿg^m=ba. However, the converse does not hold- there are abelian groups which are not cyclic, such as the one given earlier, Z₂ X Z₂. Any subgroup, factor group, product or direct sum of abelian groups will remain abelian.

By the fundamental theorem of finite abelian groups, such a group can be expressed as a direct sum of cyclic subgroups of prime-power order.

Given an abelian group, it can be shown that any subgroup will be a normal subgroup, since ∀a,h∈G, ah=ha, so the left and right cosets aH, Ha are equal. Also, for homomorphisms f,g on abelian groups, the sum f+g is also a homomorphism.