2-hypernormalized satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-hypernormalized subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

Suppose $H$ is a 2-hypernormalized subgroup of a group $G$. In other words, $N_{G}(N_{G}(H))=G$. Suppose $K$ is a subgroup of $G$ containing $H$. Then, $H$ is 2-hypernormalized in $K$: $N_{K}(N_{K}(H))=K$.

Related facts

• Finitarily hypernormalized does not satisfy intermediate subgroup condition
• 2-hypernormalized does not satisfy transfer condition

Facts used

1. Normality satisfies transfer condition

Proof

Given: $H\leq K\leq G$, $N_{G}(N_{G}(H))=G$.

To prove: $N_{K}(N_{K}(H))=K$.

Proof:

1. $N_{K}(H)=N_{G}(H)\cap K$: This follows directly from the definition of normalizer.
2. $N_{K}(H)$ is normal in $K$: By assumption, $N_{G}(H)$ is normal in $G$. Fact (1) yields that $N_{G}(H)\cap K=N_{K}(H)$ is normal in $K$.
3. $N_{K}(N_{K}(H))=K$: This follows directly from the previous step.