Group with operators
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.
A group with operators (G, ) can be defined as a group G together with an action of a set on G :
that is distributive relative 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.
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
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 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 is the set of group endomorphisms of G.
- Given any group G, (G, ∅) is trivially a group with operators
- Given an R-module M, R acts by scalar multiplication on the underlying Abelian group M, so (M, R) is a group with operators.
- As a special case of the above, every vector space over 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.
- 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.