Automorph-join-closed subnormal of normal implies conjugate-join-closed subnormal

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

Statement with symbols

If $H$ is an automorph-join-closed subnormal subgroup of $K$ and $K$ is a normal subgroup of $G$, then $H$ is a conjugate-join-closed subnormal subgroup of $G$.

Related facts

• Left residual of conjugate-join-closed subnormal by normal equals automorph-join-closed subnormal
• Finite-automorph-join-closed subnormal of normal implies finite-conjugate-join-closed subnormal
• Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal
• 3-subnormal implies finite-conjugate-join-closed subnormal

Proof

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