Group with operators

In mathematics, more specifically in abstract algebra, a group with operators or Ω-group is a group with a set of group endomorphisms.

Groups with operators are a basic concept in algebra which was extensively studied by Emmy Noether and her school in the 1920s. The three Noether isomorphism theorems hold for groups with operators and where originally formulated by Emmy Noether using this concept.

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. The elements of ${\displaystyle \Omega }$  are called homotheties of ${\displaystyle G}$ .

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

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

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

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

Notes

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

• For a group ${\displaystyle G}$  ${\displaystyle (G,\emptyset )}$  is trivially a group with operators
• Given an ${\displaystyle R}$ -module ${\displaystyle M}$  then ${\displaystyle (M,R)}$  is a group with operators, with ${\displaystyle R}$  operating on ${\displaystyle M}$  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.