# 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 $K$: $N_K(N_K(H)) = K$.

## Facts used

1. Normality satisfies transfer condition

## Proof

Given: $H \le K \le G$, $N_G(N_G(H)) = G$.

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.