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In [[abstract algebra]], a branch of pure [[mathematics]],
Groups with operators were extensively studied by [[Emmy Noether]] and her school in the [[1920]]s. She employed the concept in her original formulation of the three [[Noether isomorphism theorem]]s.
== Definition ==
A '''group with operators'''
▲A '''group with operators''' <math>(G, \Omega)</math> is a group <math>G</math> together with a family of [[function (mathematics)|function]]s <math>\Omega</math>
:<math>\omega : G \to G \quad \omega \in \Omega</math>
which are [[distributive]] with respect to the [[group operation]]. <math>\Omega</math> is called the '''operator ___domain''', and its elements are called the '''homotheties''' of
We denote the image of a group element ''g'' under a function ω with <math>g^\omega</math>. The distributivity can then be expressed as
:<math>\forall \omega \in \Omega, \forall g,h \in G \quad (gh)^{\omega} = g^{\omega}h^{\omega} .</math>
A [[subgroup]]
:<math>\forall s \in S, \forall \omega \in \Omega : s^\omega \in S.</math>
== Category theoretic remarks ==
In [[category theory]], a '''group with operators''' can be defined as an object of a [[functor category]] '''Grp'''<sup>'''M'''</sup> where '''M''' is a monoid (''i.e.'', a category with one object) and '''Grp''' denotes the [[category of groups]]. This definition is equivalent to the previous one.
A group with operators is also a mapping
:<math>\Omega\rightarrow\operatorname{End}_{\mathbf{Grp}}(G)
where
== Examples ==
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