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.

Definition

A group with operators is a group ${\displaystyle G}$  together with a set ${\displaystyle \Omega }$  and a function

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

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

Examples

• Every group together with an empty set is trivially a group with operators
• A vector space is a group with operators. The operators are the elements of the field.

References

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