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Definition

A group with operators ( G , Ω ) {\displaystyle (G,\Omega )} can be defined¹ as a group G = ( G , ⋅ ) {\displaystyle G=(G,\cdot )} together with an action of a set Ω {\displaystyle \Omega } on G {\displaystyle G} :

Ω × G → G : ( ω , g ) ↦ g ω {\displaystyle \Omega \times G\rightarrow G:(\omega ,g)\mapsto g^{\omega }}

that is distributive relative to the group law:

( g ⋅ h ) ω = g ω ⋅ h ω . {\displaystyle (g\cdot h)^{\omega }=g^{\omega }\cdot h^{\omega }.}

For each ω ∈ Ω {\displaystyle \omega \in \Omega } , the application g ↦ g ω {\displaystyle g\mapsto g^{\omega }} 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 ( u ω ) ω ∈ Ω {\displaystyle \left(u_{\omega }\right)_{\omega \in \Omega }} of endomorphisms of G.

Ω {\displaystyle \Omega } is called the operator domain. The associate endomorphisms² are called the homotheties of G.

Given two groups G, H with same operator domain Ω {\displaystyle \Omega } , a homomorphism of groups with operators from ( G , Ω ) {\displaystyle (G,\Omega )} to ( H , Ω ) {\displaystyle (H,\Omega )} is a group homomorphism ϕ : G → H {\displaystyle \phi :G\to H} satisfying

ϕ ( g ω ) = ( ϕ ( g ) ) ω {\displaystyle \phi \left(g^{\omega }\right)=(\phi (g))^{\omega }} for all ω ∈ Ω {\displaystyle \omega \in \Omega } and g ∈ G . {\displaystyle g\in G.}

A subgroup S of G is called a stable subgroup, Ω {\displaystyle \Omega } -subgroup or Ω {\displaystyle \Omega } -invariant subgroup if it respects the homotheties, that is

s ω ∈ S {\displaystyle s^{\omega }\in S} for all s ∈ S {\displaystyle s\in S} and ω ∈ Ω . {\displaystyle \omega \in \Omega .}

Category-theoretic remarks

In category theory, a group with operators can be defined³ as an object of a functor category GrpM 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, provided Ω {\displaystyle \Omega } is a monoid (if not, we may expand it to include the identity and all compositions).

A morphism in this category is a natural transformation between two functors (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 component of the natural transformation).

A group with operators is also a mapping

Ω → End G r p ⁡ ( G ) , {\displaystyle \Omega \rightarrow \operatorname {End} _{\mathbf {Grp} }(G),}

where End G r p ⁡ ( G ) {\displaystyle \operatorname {End} _{\mathbf {Grp} }(G)} is the set of group endomorphisms of G.

Examples

• Given any group G, (G, ∅) is trivially a group with operators
• Given a module M over a 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 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 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 is an operator-subgroup relative to the operator set X, of the group in question.

See also

• Group action

Notes

• Bourbaki, Nicolas (1974). Elements of Mathematics : Algebra I Chapters 1–3. Hermann. ISBN 2-7056-5675-8.
• Bourbaki, Nicolas (1998). Elements of Mathematics : Algebra I Chapters 1–3. Springer-Verlag. ISBN 3-540-64243-9.
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag. ISBN 0-387-98403-8.

References

1. Bourbaki 1974, p. 31. - Bourbaki, Nicolas (1974). Elements of Mathematics : Algebra I Chapters 1–3. Hermann. ISBN 2-7056-5675-8. https://archive.org/details/algebra0000bour ↩

2. Bourbaki 1974, pp. 30–31. - Bourbaki, Nicolas (1974). Elements of Mathematics : Algebra I Chapters 1–3. Hermann. ISBN 2-7056-5675-8. https://archive.org/details/algebra0000bour ↩

3. Mac Lane 1998, p. 41. - Mac Lane, Saunders (1998). Categories for the Working Mathematician. Springer-Verlag. ISBN 0-387-98403-8. ↩