Isomorph-containing implies characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., isomorph-containing subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
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Statement

Any isomorph-containing subgroup of a group is a characteristic subgroup.

Definitions used

Isomorph-containing subgroup

Further information: isomorph-containing subgroup

A subgroup $H$ of a group $G$ is termed an isomorph-containing subgroup if, for every subgroup $K$ of $G$ isomorphic to $H$, $K\leq H$.

Characteristic subgroup

Further information: characteristic subgroup

A subgroup $H$ of a group $G$ is termed a characteristic subgroup if, for every automorphism $\sigma$ of $G$, $\sigma (H)\leq H$.

Related facts

Stronger facts

• Isomorph-containing implies intermediately characteristic
• Isomorph-containing implies injective endomorphism-invariant
• Isomorph-containing implies intermediately injective endomorphism-invariant

Proof

Given: An isomorph-containing subgroup $H$ of a group $G$.

To prove: For every automorphism $\sigma$ of $G$, $\sigma (H)\leq H$

Proof: Since $\sigma$ is an automorphism, the restriction of $\sigma$ to $H$ is an isomorphism from $H$ to $\sigma (H)$. Hence, $\sigma (H)$ is isomorphic to $H$. Since $H$ is isomorph-containing, we get $\sigma (H)\leq H$, completing the proof.