Group with operators: Difference between revisions

Revision as of 01:42, 28 June 2021

Revision as of 18:57, 1 November 2021

Line 3:

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

{{Algebraic structures |Module}}

== Definition ==

A '''group with operators''' <math>(G, \Omega)</math> can be defined{{sfn|Bourbaki|1974|p=31}} as a group <math>G = (G, \cdot)</math> together with an action of a set <math>\Omega</math> on <math>G</math>:

: <math>\Omega \times G \rightarrow G : (\omega , g) \mapsto g^{\omega}</math>

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

: <math>(g \cdot h)^{\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 [[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''.

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

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

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

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

== Category-theoretic remarks ==

In [[category theory]], a '''group with operators''' can be defined{{sfn|Mac Lane|1998|p=41}} as an object of a [[functor category]] '''Grp'''^''M'' where ''M'' is a [[monoid]] (i.e. a [[Category (mathematics)|category]] with one [[Objectobject (category theory)|object]]) and '''Grp''' denotes the [[category of groups]]. This definition is equivalent to the previous one, provided <math>\Omega</math> is a monoid (otherwise we may expand it to include the identity and all 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''). Again we recover the definition above of a homomorphism of groups with operators (with ''f'' the [[Natural_Transformationnatural 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 endomorphisms of ''G''.

== Examples ==

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

* Given a [[module (mathematics)|module]] ''M'' over a [[Ringring (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==

The [[Jordan–Hölder theorem]] also holds in the context of operator groups. 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==