Group With Operators

A group with operators (G, ) can be defined as a group G together with an action of a set on G :

which is distributive relatively to the group law :

For each, the application

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 of endomorphisms of G.

is called the operator domain. The associate endomorphisms are called the homotheties of G.

Given two groups G, H with same operator domain, a homomorphism of groups with operators is a group homomorphism f:GH satisfying

A subgroup S of G is called a stable subgroup, -subgroup or -invariant subgroup if it respects the homotheties, that is