Difference between revisions of "Abelian group"
From Online Dictionary of Crystallography
m |
m (link) |
||
| Line 2: | Line 2: | ||
| − | 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 09:10, 29 May 2007
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.