Definition:Abelian Group

Additive Notation
When discussing abelian groups, it is customary to use additive notation, where:

Abelian Group Axioms
Under this regime, the group axioms read:

This notation gains in importance and usefulness when discussing rings.

Also known as
The usual way of spelling this is without a capital letter, that is abelian, but Abelian is frequently seen.

Also see

 * Equivalence of Definitions of Abelian Group


 * Group equals Center iff Abelian

Linguistic Note
The pronunciation of abelian is usually either a-bee-lee-an or a-bell-ee-an, putting the emphasis on the second syllable.

Note that the term abelian has thus phonetically lost the connection to its eponym (correctly pronounced aah-bl).