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

In [[mathematicsabstract algebra]], morea specificallybranch inof [[abstract algebramathematics]], a '''group with operators''' or ΩΩ-'''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 are a basic concept in algebra which waswere extensively studied by [[Emmy Noether]] and her school in the [[1920]]s1920s. TheShe threeemployed [[Noetherthe isomorphismconcept theorem]]sin holdher fororiginal groupsformulation withof operatorsthe and where originally formulated by Emmythree [[Noether usingisomorphism this concepttheorems]].

{{Algebraic structures|Module}}

== Definition ==

A '''group with operators''' <math>(G, \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:

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

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]]. The elements of <math>\Omega</math> are called '''homotheties''' of <math>G</math>.

<math>\Omega</math> is called the '''operator ___domain'''. The associate endomorphisms{{sfn|Bourbaki|1974|pp=30–31}} are called the '''homotheties''' of ''G''.

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 expresses as

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

AGiven [[subgroup]]two <math>S</math>groups of''G'', ''H'' with same operator ___domain <math>G\Omega</math>, is calleda '''stable subgrouphomomorphism''', of groups with operators from <math>(G, \Omega)</math>-'''subgroup''' orto <math>(H, \Omega)</math> '''invariantis subgroup'''a if[[group ithomomorphism]] respects<math>\phi: theG hometheties,\to thatH</math> issatisfying

: <math>\forallphi\left(g^\omega\right) s= (\inphi(g))^\omega</math> S,for \forallall <math>\omega \in \Omega</math> :and s^\omega<math>g \in SG.</math>

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

== Notes ==

: <math>s^\omega \in S</math> for all <math>s \in S</math> and <math>\omega \in \Omega.</math>

== Category-theoretic remarks ==

A group with operators is a mapping

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).

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

A group with operators is also a mapping

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

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

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

== Examples ==

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

* Given a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'', ''R'' acts by [[scalar multiplication]] on the underlying [[abelian group]] 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==

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

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.

* 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.

==See References also==

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

== Notes ==

*{{cite book | author=Bourbaki, Nicolas | title=Elements of Mathematics : Algebra I Chapters 1-3 | publisher=Springer-Verlag | year=1998 | id=ISBN 3540642439}}

{{reflist}}

== References ==

*{{cite book | last=Bourbaki | first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1–3 | publisher=Hermann | year=1974 | isbn=2-7056-5675-8 | url-access=registration | url=https://archive.org/details/algebra0000bour }}

*{{cite book | authorlast=Bourbaki, | first=Nicolas | title=Elements of Mathematics : Algebra I Chapters 1-31–3 | publisher=Springer-Verlag | year=1998 | idisbn=ISBN 35406424393-540-64243-9}}

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

[[Category:AbstractGroup algebraactions]]
[[Category:Universal algebra]]