Group with operators: Difference between revisions

Revision as of 21:46, 13 September 2006

Revision as of 21:50, 13 September 2006

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== Definition ==

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

We denote the image of a group element ''g'' under a function ω<math>\omega</math> 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]] ''S'' of ''G'' is called a '''stable subgroup''', ω-'''<math>\omega</math>-subgroup''' or Ω-'''<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>