Group with operators
A group with operators is a group with a set of operators
such that each
is associated with a group
endomorphism
on
.
By abuse of notation, we usually refer
to as simply
, and we write
as
,
when
is written multiplicatively; when
is written additively,
we usually write
, or simply
.
A subgroup of a group with operators is called a stable subgroup if it is closed under the action of the operators. It is called a normal stable subgroup if it is a normal subgroup and a stable subgroup.
In practice, we deal with general groups with operators infrequently. However, many structures&emdash;groups and modules (including rings, fields, and vector spaces)&emdash;are special cases of this general structure, and we can prove many results&emdash;for example, the Jordan-Hölder Theorem&emdash;about groups with operators in general; we then avoid repeated proofs of the same results in different fields.
Emmy Noether was responsible for much of the study of groups with operators.
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