Group with operators: Difference between revisions

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In [[mathematicsabstract algebra]], morea specificallybranch inof pure [[abstract algebramathematics]], a '''group with operators''' or Ω-'''group''' is a [[group (mathematics)|group]] with a [[set (mathematics)|set]] of group [[endomorphism]]s.

Groups with operators are a basic concept in algebra which waswere extensively studied by [[Emmy Noether]] and her school in the [[1920]]s. TheShe threeemployed [[Noetherthe isomorphismconcept theorem]]sin holdher fororiginal groupsformulation withof operatorsthe and where originally formulated by Emmythree [[Noether usingisomorphism this concepttheorem]]s.

== Definition ==

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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 the '''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>(\forall \omega \in \Omega, \forall g,h \circin hG \quad (gh)^{\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 a '''stable subgroup''', <math>\Omega</math>-'''subgroup''' or <math>\Omega</math> '''invariant subgroup''' if it respects the hometheties, that is

Using [[category theory]], a '''group with operators''' can be defined as an object of a [[functor category]] '''Grp'''^'''M''' where '''M''' is a monoid (''i.e.'', a category with one object) and '''Grp''' denotes the category of groups. This definition is equivalent to the previous one.

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>

== Category theoretic remarks ==

== Notes ==

UsingIn [[category theory]], a '''group with operators''' can be defined as an object of a [[functor category]] '''Grp'''^'''M''' where '''M''' is a monoid (''i.e.'', a category with one object) and '''Grp''' denotes the category of groups. This definition is equivalent to the previous one.

A group with operators is a mapping

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

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== Examples ==

* ForGiven aany group <math>''G</math>'', <math>(''G'', \emptyset&empty;)</math> is trivially a group with operators

* Given an <math>''R</math>''-[[module (mathematics)|module]] <math>M</math> then <math>(''M'', R)</math> is athe group with''R'' operators,operates withon <math>R</math>the operatingoperator on___domain <math>''M</math>'' by [[scalar multiplication]]. More concretely, every [[vector space]] is a group with operators.

* For a group <math>G</math> <math>(G, \emptyset)</math> is trivially a group with operators

* Given an <math>R</math>-[[module (mathematics)|module]] <math>M</math> then <math>(M, R)</math> is a group with operators, with <math>R</math> operating on <math>M</math> 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 3540642439}}