Group with operators: Difference between revisions

Revision as of 09:43, 26 July 2006

Revision as of 21:27, 29 August 2006

Line 1:

In [[abstract algebra]], a branch of pure [[mathematics]], the [[algebraic structure]] '''group with operators''' or ΩΩ-'''group''' is a [[group (mathematics)|group]] with a [[set (mathematics)|set]] of group [[endomorphism]]s.

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''' (''G'', ωω) is a group ''G'' together with a family of [[function (mathematics)|function]]s ΩΩ:

:<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 &omega;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]] ''S'' of ''G'' is called a '''stable subgroup''', &omega;ω-'''subgroup''' or &Omega;Ω-'''invariant subgroup''' if it respects the hometheties, that is

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

Line 23:

== Examples ==

* Given any group ''G'', (''G'', &empty;∅) is trivially a group with operators

* Given an ''R''-[[module (mathematics)|module]] ''M'', the group ''R'' operates on the operator ___domain ''M'' by [[scalar multiplication]]. More concretely, every [[vector space]] is a group with operators.

== References ==

*{{cite book | author=Bourbaki, Nicolas | title=Elements of Mathematics : Algebra I Chapters 1-3 | publisher=Springer-Verlag | year=1998 | id=ISBN 35406424393-540-64243-9}}

[[Category:Abstract algebra]]
[[Category:Universal algebra]]