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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 '''operator ___domain''' and its elements are called '''homotheties''' of <math>G.</math>
We denote the image of a group element <math>g</math> under a function <math>\omega</math> with <math>g^\omega</math>. The distributivity can then be expressed as
:<math>(g \circ h)^{\omega} = g^{\omega} \circ h^{\omega} \quad \forall \omega \in \Omega, \forall g,h \in G.</math>
A [[subgroup]] <math>S</math> of <math>G</math> is called '''stable subgroup''', <math>\Omega</math>-'''subgroup''' or <math>\Omega</math> '''invariant subgroup''' if it respects the hometheties, that is
:<math>\forall s \in S, \forall \omega \in \Omega : s^\omega \in S.</math>
== Notes ==
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