Automorph-permutable of normal implies conjugate-permutable

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Automorph-permutable subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Conjugate-permutable subgroup (?))
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Statement

Property-theoretic statement

Automorph-permutable * Normal $\leq$ Normal

here $*$ denotes the composition operator.

Verbal statement

Any automorph-permutable subgroup of a normal subgroup is conjugate-permutable.

Symbolic statement

Suppose $H\leq K\leq G$ are groups, such that $H$ is automorph-permutable in $K$, and $K$ is normal in $G$. Then $H$ is conjugate-permutable in $G$.

Related facts

• Characteristic of normal implies normal: Both proofs use very similar arguments.
• 2-subnormal implies conjugate-permutable: This statement follows from the statement of this article, using the fact that normal implies automorph-permutable.

Proof

Suppose $H\leq K\leq G$ are groups, such that $H$ is automorph-permutable in $K$, and $K$ is normal in $G$. We need to show that $H$ is conjugate-permutable in $G$.

We do this as follows. First, pick $g\in G$, and consider the operation $c_{g}:x\mapsto gxg^{-1}$. We need to show that $H$ and $c_{g}(H)=H^{g}=gHg^{-1}$ permute. Since $K$ is a normal subgroup of $G$, $c_{g}$ restricts to an automorphism of $K$, and thus $c_{g}(H)$ (simply $gHg^{-1}$ is an automorph of $H$ in $K$). Since $H$ is automorph-permutable inside $K$, $H$ permutes with $c_{g}(H)$, completing the proof.