Group with operators

Agroup 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 endomorphismofG. 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 homothetiesofG.

Given two groups G, H with same operator domain ${\displaystyle \Omega }$ , a homomorphism of groups with operators from ${\displaystyle (G,\Omega )}$ to${\displaystyle (H,\Omega )}$  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 ${\displaystyle g\in G.}$ 

Asubgroup SofG is called a stable subgroup, ${\displaystyle \Omega }$ -subgroupor${\displaystyle \Omega }$ -invariant subgroup if it respects the homotheties, that is

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

Incategory 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 (if not, we may expand it to include the identity and all compositions).

Amorphism 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 groupofM, 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 groups with operators. The requirement that a group have a composition series is analogous to that of compactnessintopology, 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.