In abstract algebra, a branch of pure 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.
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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 (G,
that is distributive relatively to the group law :
For each
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
Given two groups G, H with same operator domain
A subgroup S of G is called a stable subgroup,
Category-theoretic remarks
In category theory, a group with operators can be defined 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
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
Examples
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.