Group with operators

A group with operators ${\displaystyle (G,\Omega )}$ can be defined[1] as a group ${\displaystyle G=(G,\cdot )}$ together with an action of a set ${\displaystyle \Omega }$ on ${\displaystyle G}$:

${\displaystyle \Omega \times G\rightarrow G:(\omega ,g)\mapsto g^{\omega }}$

that is distributive relative to the group law:

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

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

${\displaystyle \Omega }$ is called the operator domain. The associate endomorphisms[2] are called the homotheties of G.

Given two groups G, H with same operator domain ${\displaystyle \Omega }$, a homomorphism of groups with operators is a group homomorphism ${\displaystyle \phi :G\to H}$ satisfying

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

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

${\displaystyle s^{\omega }\in S}$ for all ${\displaystyle s\in S}$ and ${\displaystyle \omega \in \Omega .}$

In category theory, a group with operators can be defined[3] 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, provided ${\displaystyle \Omega }$ 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 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

${\displaystyle \Omega \rightarrow \operatorname {End} _{\mathbf {Grp} }(G),}$

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

• 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).

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 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.

• Group action

• 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.