Normal-isomorph-automorphic subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed a normal-isomorph-automorphic subgroup if $H$ is a normal subgroup of $G$ and for any normal subgroup $K$ of $G$ isomorphic to $H$, $H$ and $K$ are automorphic subgroups in $G$, i.e., there is an automorphism $\sigma$ of $G$ such that $\sigma (H)=K$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal-isomorph-free subgroup	normal, and no other isomorphic normal subgroup			|FULL LIST, MORE INFO
Isomorph-free subgroup	No other isomorphic subgroup	(via normal-isomorph-free)	(via normal-isomorph-free)	|FULL LIST, MORE INFO
Isomorph-automorphic normal subgroup	Normal, and automorphic to all isomorphic subgroups			|FULL LIST, MORE INFO