Group with operators

A group with operators is a group $G$ with a set of operators $\Omega$ such that each $\alpha \in \Omega$ is associated with a group endomorphism $f_\alpha$ on $G$.

By abuse of notation, we usually refer to $f_\alpha$ as simply $\alpha$, and we write $f_\alpha(g)$ as $g^{\alpha}$, when $G$ is written multiplicatively; when $G$ is written additively, we usually write $\alpha(g)$, or simply $\alpha g$.

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—groups and modules (including rings, fields, and vector spaces)—are special cases of this general structure, and we can prove many results—for example, the Jordan-Hölder Theorem—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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