# Isomorph-normal coprime automorphism-invariant subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: isomorph-normal subgroup and coprime automorphism-invariant subgroup
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## Definition

A subgroup $H$ of a finite group $G$ is termed isomorph-normal coprime automorphism-invariant if it satisfies the following two conditions:

• $H$ is isomorph-normal in $G$: Every subgroup of $G$ isomorphic to $H$ is a normal subgroup of $G$.
• $H$ is a coprime automorphism-invariant subgroup of $G$: Any automorphism $\sigma$ of $G$ whose order is relatively prime to the order of $G$ satisfies $\sigma(H) = H$.