Group with operators: Difference between revisions

Revision as of 14:16, 5 March 2006

Revision as of 16:43, 14 April 2006

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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 expressesexpressed 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>