Group with operators: Difference between revisions

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In [[abstract algebra]], a branch of pure [[mathematics]], the [[algebraic structure]] '''group with operators''' or Ω-'''group''' is a [[group (mathematics)|group]] with a [[setfamily (mathematics)|setfamily]] of group [[endomorphism]]s.

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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== Definition ==

A '''group with operators''' (''G'', <math>\Omega</math>) can be defined{{sfn|Bourbaki|1974|p=31}} as a group ''G'' together with aan familyaction of [[functiona (mathematics)|function]]sset <math>\Omega</math> on ''G'' :

:<math>\omega :\Omega \times G \torightarrow G \quad: (\omega , g) \inmapsto g^{\Omegaomega}</math>

which is distributive relatively to the group law :

which are [[distributive]] with respect to the [[group operation]]. <math>\Omega</math> is called the '''operator ___domain''', and its elements are [[endomorphisms]]{{sfn|Bourbaki|1974|pp=30-31}} called the '''homotheties''' of ''G''.

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

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

:<math>\ g \mapsto g^{\omega}</math>

is then an endomorphism of ''G''. From this, it results that a Ω-group can also be viewed as a group ''G'' with a family <math>(u_{\omega})_{\omega \in \Omega}</math> of endomorphisms of ''G''.

which are [[distributive]] with respect to the [[group operation]]. <math>\Omega</math> is called the '''operator ___domain''',. andThe its elements areassociate [[endomorphisms]]{{sfn|Bourbaki|1974|pp=30-31}} are called the '''homotheties''' of ''G''.

We denote the image of a group element ''g'' under a function <math>\omega</math> with <math>g^\omega</math>. The distributivity can then be expressed as