# Difference between revisions of "Abelian group"

### From Online Dictionary of Crystallography

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− | <font color="blue">Groupe abélien</font> (''Fr'') | + | <font color="blue">Groupe abélien</font> (''Fr''). <font color="red">Abelsche Gruppe</font> (''Ge''). <font color="green">Grupo abeliano</font> (''Sp''). <font color="black">Gruppo abeliano</font> (''It''). <font color="purple">アーベル群</font> (''Ja''). |

− | + | 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. | ||

Groups that are not commutative are called non-abelian (rather than non-commutative). | Groups that are not commutative are called non-abelian (rather than non-commutative). |

## Revision as of 10:09, 12 May 2017

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 *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.