Nilpotent join of pronormal subgroups is pronormal

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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.

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 \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.