Group with operators: Difference between revisions

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== Definition ==

A '''group with operators''' is a group <math>G</math> together with a setfamily of [[function (mathematics)|function]]s <math>\Omega</math> and a function

:<math>\omega : G \to G \quad \omega \in \Omega</math>

which are [[distributive]] with respect to the [[group operation]]. The elements of <math>\Omega</math> 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 expresses as

:<math>\Omega\rightarrow\operatorname{End}_{\mathbf{Grp}}(G)</math>,

:<math>(g \circ h)^{\omega} = g^{\omega} \circ h^{\omega} \quad \forall \omega \in \Omega, \forall g,h \in G</math>.

where <math>\mathbf{Grp}</math> is theA [[category of groupssubgroup]] and <math>\operatorname{End}_{\mathbf{Grp}}(G)S</math> is the set of group [[endomorphism]]s of <math>G</math>. Theis elementscalled of'''stable subgroup''' or <math>\Omega</math> are called -'''homothetiessubgroup'''. if it respects the hometheties, that is

:<math>\forall s \in S, \forall \omega \in \Omega : s^\omega \in S</math>

== Notes ==

A group with operators is a mapping

:<math>\Omega\rightarrow\operatorname{End}_{\mathbf{Grp}}(G)</math>,

where <math>\mathbf{Grp}</math> is the [[category of groups]] and <math>\operatorname{End}_{\mathbf{Grp}}(G)</math> is the set of group [[endomorphism]]s of <math>G</math>.

== Examples ==