Subnormal-to-normal is normalizer-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal-to-normal subgroup) satisfying a subgroup metaproperty (i.e., normalizer-closed subgroup property)
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Statement

Suppose $H$ is a subnormal-to-normal subgroup of a group $G$: either $H$ is normal in $G$ or $H$ is not subnormal in $G$. Then, the normalizer of $H$ in $G$ is also a subnormal-to-normal subgroup of $G$.

Related facts

• Intermediately subnormal-to-normal is normalizer-closed
• Intermediately normal-to-characteristic is normalizer-closed
• Automorph-conjugacy is normalizer-closed
• Intermediate automorph-conjugacy is normalizer-closed

Proof

Given: A subnormal-to-normal subgroup $H$ of a group $G$, with normalizer $N_{G}(H)$.

To prove: $N_{G}(H)$ is also a subnormal-to-normal subgroup of $G$.

Proof: If $N_{G}(H)$ is not subnormal in $G$ we are done. So, we need to prove that if $N_{G}(H)$ is a subnormal subgroup of $G$, $N_{G}(H)$ is normal in $G$.

If $N_{G}(H)$ is subnormal in $G$, then, since $H$ is normal in $N_{G}(H)$, $H$ is subnormal in $G$. Since $H$ is subnormal-to-normal, $H$ is in fact normal in $G$, so $N_{G}(H)=G$. Since every group is normal as a subgroup of itself, $N_{G}(H)$ is a normal subgroup of $G$.