Group with operators: Difference between revisions

Revision as of 07:47, 15 March 2013

Revision as of 17:00, 23 November 2014

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In [[abstract algebra]], a branch of pure [[mathematics]], the [[algebraic structure]] '''group with operators''' or Ω-'''group''' can be viewed as a [[group (mathematics)|group]] with a [[Set (mathematics)|set]] Ω whichthat operates on the elements of the group in a special way.

Groups with operators were extensively studied by [[Emmy Noether]] and her school in the 1920s. She employed the concept in her original formulation of the three [[Noether isomorphism theorem]]s.

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A '''group with operators''' (''G'', <math>\Omega</math>) can be defined{{sfn|Bourbaki|1974|p=31}} as a group ''G'' together with an action of a set <math>\Omega</math> on ''G'' :

:<math>\ \Omega \times G \rightarrow G : (\omega , g) \mapsto g^{\omega}</math>

whichthat is distributive relatively to the group law :

:<math>\ (gh)^{\omega} = g^{\omega} h^{\omega}.</math>

For each <math>\omega \in \Omega </math>, the application