Group with operators

In abstract algebra, a branch of mathematics, a group with operators or Ω-group is an algebraic structure that can be viewed as a group together with a set Ω that operates on the elements of the group in a special way.

Groups with operators were extensively studied by

Definition

A group 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 endomorphism of G. 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 homotheties of G.

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 ${\displaystyle g\in G.}$

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

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

Category-theoretic remarks

In

. This definition is equivalent to the previous one, provided ${\displaystyle \Omega }$ is a monoid (if not, we may expand it to include the

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

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

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

Examples

• 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 group of M, 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).

Applications

The

References

• 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
  .