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{{Short description|Concept in mathematics regarding sets operating on groups}}

In [[abstract algebra]], a branch of pure [[mathematics]], the [[algebraic structure]]a '''group with operators''' or Ω-'''group''' is an [[algebraic structure]] that can be viewed as a [[group (mathematics)|group]] together with a [[set (mathematics)|set]] Ω that operates on the elements of the group [[endomorphism]]sin a special way.

Groups with operators were extensively studied by [[Emmy Noether]] and her school in the [[1920]]s1920s. She employed the concept in her original formulation of the three [[Noether isomorphism theoremtheorems]]s.

{{Algebraic structures|Module}}

== Definition ==

A '''group with operators''' <math>(''G'', <math>\Omega)</math>) iscan be defined{{sfn|Bourbaki|1974|p=31}} as a group ''<math>G'' = (G, \cdot)</math> together with aan familyaction of [[functiona (mathematics)|function]]sset <math>\Omega</math> on <math>G</math>:

: <math>\omegaOmega :\times G \torightarrow G \quad: (\omega, g) \inmapsto g^\Omegaomega</math>

that is [[distributive property|distributive]] relative to the group law:

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''.

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

For each <math>\omega \in \Omega </math>, the application <math>g \mapsto g^\omega</math> is then an [[Group homomorphism#Types|endomorphism]] of ''G''. From this, it results that a Ω-group can also be viewed as a group ''G'' with an [[indexed family]] <math>\left(u_\omega\right)_{\omega \in \Omega}</math> of endomorphisms of ''G''.

We denote the image of a group element ''g'' 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 \in G \quad (gh)^{\omega} = g^{\omega}h^{\omega} .</math>

which are [[distributive]] with respect to the [[group operation]]. <math>\Omega</math> is called the '''operator ___domain''',. andThe itsassociate elementsendomorphisms{{sfn|Bourbaki|1974|pp=30–31}} are called the '''homotheties''' of ''G''.

A [[subgroup]] ''S'' of ''G'' is called a '''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>

Given two groups ''G'', ''H'' with same operator ___domain <math>\Omega</math>, a '''homomorphism''' of groups with operators from <math>(G, \Omega)</math> to <math>(H, \Omega)</math> is a [[group homomorphism]] <math>\phi: G \to H</math> satisfying

== Category theoretic remarks ==

: <math>\phi\left(g^\omega\right) = (\phi(g))^\omega</math> for all <math>\omega \in \Omega</math> and <math>g \in G.</math>

In [[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]] ''S'' of ''G'' is called a '''stable subgroup''', '''<math>\omegaOmega</math>-subgroup''' or '''<math>\Omega</math>-invariant subgroup''' if it respects the homethetieshomotheties, that is

: <math>s^\forall somega \in S,</math> \forallfor \omegaall <math>s \in \OmegaS</math> :and s^<math>\omega \in S\Omega.</math>

== Category -theoretic remarks ==

In [[category theory]], a '''group with operators''' can be defined{{sfn|Mac Lane|1998|p=41}} as an [[object (category theory)|object]] of a [[functor category]] '''Grp'''^'''M''' where '''M''' is a [[monoid]] (''i.e.'', a [[category (mathematics)|category]] with one object) and '''Grp''' denotes the [[category of groups]]. This definition is equivalent to the previous one, provided <math>\Omega</math> is a monoid (if not, we may expand it to include the [[identity function|identity]] and all [[function composition|compositions]]).

A [[morphism]] in this category is a [[natural transformation]] between two [[functor]]s (i.e., two groups with operators sharing same operator ___domain ''M''{{hairsp}}). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the [[natural transformation#Definition|component]] of the natural transformation).

A group with operators is also a mapping

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

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

== Examples ==

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

* Given ana ''R''-[[module (mathematics)|module]] ''M'', theover groupa [[ring (mathematics)|ring]] ''R'' operates on the operator ___domain, ''MR'' acts by [[scalar multiplication]]. Moreon concretely,the everyunderlying [[vectorabelian spacegroup]] of ''M'', so (''M'', ''R'') is a group with operators.

* As a special case of the above, every [[vector space]] over a [[field (mathematics)|field]] ''K'' is a group with operators (''V'', ''K'').

==Applications==

The [[Jordan–Hölder theorem]] also holds in the context of groups with operators. The requirement that a group have a [[composition series]] is analogous to that of [[compact space|compactness]] in [[topology]], and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about composition series where each ([[Normal subgroup|normal]]) subgroup is an operator-subgroup relative to the operator set ''X'', of the group in question.

==See also==

* [[Group action (mathematics)|Group action]]

==Notes==

{{reflist}}

== References ==

*{{cite book | authorlast=Bourbaki, Nicolas| first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1-31–3 | publisher=Springer-VerlagHermann | year=19981974 | idisbn=ISBN 32-5407056-642435675-98 | url-access=registration | url=https://archive.org/details/algebra0000bour }}

*{{cite book | last=Bourbaki | first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1–3 | publisher=Springer-Verlag | year=1998 | isbn=3-540-64243-9}}

*{{cite book | last=Mac Lane | first=Saunders | title=Categories for the Working Mathematician | publisher=Springer-Verlag | year=1998 | isbn=0-387-98403-8}}

[[Category:Group actions]]