Editing Composition series (section)

==Generalization==
[[group with operators|Groups with a set of operators]] generalize group actions and ring actions on a group. A unified approach to both groups and modules can be followed as in {{harv|Bourbaki|1974|loc=Ch. 1}} or {{harv|Isaacs|1994|loc=Ch. 10}}, simplifying some of the exposition. The group ''G'' is viewed as being acted upon by elements (operators) from a set Ω. Attention is restricted entirely to subgroups invariant under the action of elements from Ω, called Ω-subgroups. Thus Ω-composition series must use only Ω-subgroups, and Ω-composition factors need only be Ω-simple. The standard results above, such as the Jordan–Hölder theorem, are established with nearly identical proofs.

The special cases recovered include when Ω = ''G'' so that ''G'' is acting on itself. An important example of this is when elements of ''G'' act by conjugation, so that the set of operators consists of the [[inner automorphism]]s. A composition series under this action is exactly a [[chief series]]. Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.