Glossary of Group Theory - Abelian Groups

Abelian Groups

The category of groups can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian (in honor of Niels Abel, or commutative) groups, i.e. the ones satisfying

Another way of saying this is that the commutator

equals the identity element. A non-abelian group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. Being either isomorphic to Z or to Z_n, the integers modulo n, they are always abelian. Any finitely generated abelian group is known to be a direct sum of groups of these two types. The category of abelian groups is an abelian category. In fact, abelian groups serve as the prototype of abelian categories. A converse is given by Mitchell's embedding theorem.