Nilpotent join of pronormal subgroups is pronormal

Statement

Suppose $H,K\leq G$ are Pronormal subgroup (?)s, such that $\langle H,K\rangle$ is a Nilpotent group (?). Then:

• $H$ and $K$ normalize each other.
• $\langle H,K\rangle =HK$ is also a pronormal subgroup of $G$.

Facts used

1. Pronormality satisfies intermediate subgroup condition
2. Nilpotent implies every subgroup is subnormal
3. Pronormal and subnormal implies normal
4. Pronormality is normalizing join-closed

Proof

Given: $H,K\leq G$ are pronormal subgroups, and $\langle H,K\rangle$ is nilpotent.

To prove: $H$ and $K$ normalize each other, and $HK=\langle H,K\rangle$ is also pronormal.

Proof: By fact (1), $H,K$ are both pronormal in $\langle H,K\rangle$. By fact (2), they are also both subnormal in $\langle H,K\rangle$. By fact (3), $H,K$ are both normal in $\langle H,K$. Thus, we have $HK=\langle H,K\rangle$, and that $H$ and $K$ normalize each other. By fact (4), we obtain that $\langle H,K\rangle =HK$ is also pronormal.