Nilpotent join of pronormal subgroups is pronormal

Statement

Suppose $H, K \le G$ 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 $G$.

Proof

Given: $H, K \le G$ 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.