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&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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Group with operators

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