group with operators

In abstract algebra, a branch of mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way. 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 can be defined as a group together with an action of a set on : that is distributive relative to the group law: For each , the application 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 of endomorphisms of G.