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Group with operators

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.

Agroup with operators ${\displaystyle (G,\Omega )}$ can be defined^[1] as a group ${\displaystyle G=(G,\cdot )}$ together with an action of a set ${\displaystyle \Omega }$on${\displaystyle G}$:

${\displaystyle \Omega \times G\rightarrow G:(\omega ,g)\mapsto g^{\omega }}$

that is distributive relative to the group law:

${\displaystyle (g\cdot h)^{\omega }=g^{\omega }\cdot h^{\omega }.}$

For each ${\displaystyle \omega \in \Omega }$, the application ${\displaystyle g\mapsto g^{\omega }}$ is then an endomorphismofG. From this, it results that a Ω-group can also be viewed as a group G with an indexed family ${\displaystyle \left(u_{\omega }\right)_{\omega \in \Omega }}$ of endomorphisms of G.

${\displaystyle \Omega }$ is called the operator domain. The associate endomorphisms^[2] are called the homothetiesofG.

Given two groups G, H with same operator domain ${\displaystyle \Omega }$, a homomorphism of groups with operators from ${\displaystyle (G,\Omega )}$to${\displaystyle (H,\Omega )}$ is a group homomorphism ${\displaystyle \phi :G\to H}$ satisfying

${\displaystyle \phi \left(g^{\omega }\right)=(\phi ($g))^{\omega }} for all ${\displaystyle \omega \in \Omega }$ and ${\displaystyle g\in G.}$

Asubgroup SofG is called a stable subgroup, ${\displaystyle \Omega }$-subgroupor${\displaystyle \Omega }$-invariant subgroup if it respects the homotheties, that is

${\displaystyle s^{\omega }\in S}$ for all ${\displaystyle s\in S}$ and ${\displaystyle \omega \in \Omega .}$

Incategory theory, a group with operators can be defined^[3] 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 ${\displaystyle \Omega }$ is a monoid (if not, we may expand it to include the identity and all compositions).

Amorphism 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

${\displaystyle \Omega \rightarrow \operatorname {End} _{\mathbf {Grp} }($G),}

where ${\displaystyle \operatorname {End} _{\mathbf {Grp} }($G)} is the set of group endomorphisms of G.

• Given any group G, (G, ∅) is trivially a group with operators
• Given a module M over a ring R, R acts by scalar multiplication on the underlying abelian groupofM, so (M, R) is a group with operators.
• As a special case of the above, every vector space over a field K is a group with operators (V, K).

The Jordan–Hölder theorem also holds in the context of groups with operators. The requirement that a group have a composition series is analogous to that of compactnessintopology, 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.

• Group action

• 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.