# Action-isomorph-free subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed action-isomorph-free in $G$ if $H$ is a normal subgroup of $G$, and the following condition holds.

Suppose $\alpha:G \to \operatorname{Aut}(H)$ be the homomorphism induced by the action of $G$ on $H$ by conjugation. Suppose $K$ is a normal subgroup of $G$ with $\beta:G \to \operatorname{Aut}(K)$ the homomorphism induced by the action of $G$ on $K$ by conjugation. Suppose, further, that $\sigma:H \to K$ is an isomorphism with the property that: