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 Zn, 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.
Read more about this topic: Glossary Of Group Theory
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