From Online Dictionary of Crystallography
Revision as of 10:09, 12 May 2017 by BrianMcMahon (Style edits to align with printed edition)
Groupe abélien (Fr). Abelsche Gruppe (Ge). Grupo abeliano (Sp). Gruppo abeliano (It). アーベル群 (Ja).
An abelian group, also called a commutative group, is a group (G, * ) such that g1 * g2 = g2 * g1 for all g1 and g2 in G, where * is a binary operation in G. This means that the order in which the binary operation is performed does not matter, and any two elements of the group commute.
Groups that are not commutative are called non-abelian (rather than non-commutative).
Abelian groups are named after Niels Henrik Abel.