Group with operators

In abstract algebra, a branch of pure mathematics, a group with operators or Ω-group is a group with a set of group endomorphisms.

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 ${\displaystyle (G,\Omega )}$  is a group ${\displaystyle G}$  together with a family of functions ${\displaystyle \Omega }$ 

${\displaystyle \omega :G\to G\quad \omega \in \Omega }$ 

which are distributive with respect to the group operation. ${\displaystyle \Omega }$  is called the operator ___domain, and its elements are called homotheties of ${\displaystyle G.}$ 

We denote the image of a group element g under a function ω with ${\displaystyle g^{\omega }}$ . The distributivity can then be expressed as

${\displaystyle \forall \omega \in \Omega ,\forall g,h\in G\quad (gh)^{\omega }=g^{\omega }h^{\omega }.}$ 

A subgroup ${\displaystyle S}$  of ${\displaystyle G}$  is called a stable subgroup, ${\displaystyle \Omega }$ -subgroup or ${\displaystyle \Omega }$  invariant subgroup if it respects the hometheties, that is

${\displaystyle \forall s\in S,\forall \omega \in \Omega :s^{\omega }\in S.}$ 

Category theoretic remarks

In category theory, a group with operators can be defined 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.

A group with operators is a mapping

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

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

Examples

• Given any group G, (G, ∅) is trivially a group with operators
• Given an R-module M, the group R operates on the operator ___domain M by scalar multiplication. More concretely, every vector space is a group with operators.

References

• Bourbaki, Nicolas (1998). Elements of Mathematics : Algebra I Chapters 1-3. Springer-Verlag. ISBN 3540642439.