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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 which 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.

Template:Algebraic structures

Definition

A group with operators (G, ) can be definedTemplate:Sfn 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 endomorphismsTemplate:Sfn 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

Category-theoretic remarks

In category theory, a group with operators can be definedTemplate:Sfn 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.

Examples

  • 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).

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.

See also

Notes

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References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534