Group with operators

In abstract algebra, a branch of mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way.
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 theorems. 
Definition.
A group with operators formula_1 can be defined as a group formula_2 together with an action of a set formula_3 on formula_4:
that is distributive relative to the group law:
For each formula_7, the application formula_8 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 formula_9 of endomorphisms of "G".
formula_3 is called the operator domain. The associate endomorphisms are called the homotheties of "G".
Given two groups "G", "H" with same operator domain formula_3, a homomorphism of groups with operators is a group homomorphism formula_12 satisfying
A subgroup "S" of "G" is called a stable subgroup, formula_16-subgroup or formula_3-invariant subgroup if it respects the homotheties, that is
Category-theoretic remarks.
In category theory, a group with operators can be defined 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 formula_3 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
where formula_23 is the set of group endomorphisms of "G".
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 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.