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
Famous quotes containing the word groups:
“If we can learn ... to look at the ways in which various groups appropriate and use the mass-produced art of our culture ... we may well begin to understand that although the ideological power of contemporary cultural forms is enormous, indeed sometimes even frightening, that power is not yet all-pervasive, totally vigilant, or complete.”
—Janice A. Radway (b. 1949)