Knowlpedia

Abelian Group

A group whose operation is commutative.

An abelian group is a (G,⋅)(G,\cdot) such that, for all a,b∈Ga,b\in G,

a⋅b=b⋅a.a\cdot b = b\cdot a.
Notation

Abelian groups are often written additively: the operation is ++, the identity is 00, and the inverse of aa is −a-a.

Examples
  • (Z,+)(\mathbb Z,+) and (Z/nZ,+)(\mathbb Z/n\mathbb Z,+) are abelian groups.
  • Every is an abelian group under addition.
  • The symmetric group S3S_3 is not abelian.