Definition:Abelian Group/Definition 1

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An abelian group is a group $G$ where:

$\forall a, b \in G: a b = b a$

That is, every element in $G$ commutes with every other element in $G$.

Also known as

The usual way of spelling abelian group is without a capital letter, but Abelian is frequently seen.

The term commutative group can occasionally be seen.

Also see

Source of Name

This entry was named for Niels Henrik Abel.

Historical Note

The importance of abelian groups was discovered by Niels Henrik Abel during the course of his investigations into the theory of equations.

Linguistic Note

The pronunciation of abelian in the term abelian group 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 Abel (correctly pronounced aah-bl).