Action-isomorph-free subgroup

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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:

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