Abelian group

Written by Nate Soares, Qiaochu_Yuan, et al. last updated

An abelian group is a group where is commutative. In other words, the group operation satisfies the five axioms:

  1. Closure: For all in , is defined and in . We abbreviate as .
  2. Associativity: for all in .
  3. Identity: There is an element such that for all in , .
  4. Inverses: For each in is an element in such that .
  5. Commutativity: For all in , .

The first four are the standard group axioms; the fifth is what distinguishes abelian groups from groups.

Commutativity gives us license to re-arrange chains of elements in formulas about commutative groups. For example, if in a commutative group with elements , we have the claim , we can shuffle the elements to get and reduce this to the claim . This would be invalid for a nonabelian group, because doesn't necessarily equal in general.

Abelian groups are very well-behaved groups, and they are often much easier to deal with than their non-commutative counterparts. For example, every Subgroup of an abelian group is normal, and all finitely generated abelian groups are a direct product of cyclic groups (the structure theorem for finitely generated abelian groups).