# Abelian group

### From Online Dictionary of Crystallography

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- | <font color="orange">زمرة أبيلية</font> (''Ar'') | + | <font color="orange">زمرة أبيلية</font> (''Ar''). <font color="blue">Groupe abélien</font> (''Fr''). <font color="red">Abelsche Gruppe</font> (''Ge''). <font color="black">Gruppo abeliano</font> (''It''). <font color="purple">アーベル群</font> (''Ja''). <font color="brown">Абелева группа</font> (''Ru''). <font color="green">Grupo abeliano</font> (''Sp''). |

An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> 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. | An '''abelian group''', also called a ''commutative group'', is a group (''G'', * ) such that ''g''<sub>1</sub> * ''g''<sub>2</sub> = ''g''<sub>2</sub> * ''g''<sub>1</sub> for all ''g''<sub>1</sub> and ''g''<sub>2</sub> 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. |

## Revision as of 17:27, 8 November 2017

زمرة أبيلية (*Ar*). Groupe abélien (*Fr*). Abelsche Gruppe (*Ge*). Gruppo abeliano (*It*). アーベル群 (*Ja*). Абелева группа (*Ru*). Grupo abeliano (*Sp*).

An **abelian group**, also called a *commutative group*, is a group (*G*, * ) such that *g*_{1} * *g*_{2} = *g*_{2} * *g*_{1} for all *g*_{1} and *g*_{2} 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.