Pronormality is not permuting join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., permuting join-closed subgroup property).
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Statement

It is possible to have a group $G$ and two permuting subgroups $H,K$ of $G$ such that both $H$ and $K$ are pronormal subgroups but the join $\langle H,K\rangle$, which in this case equals the product $HK$, is not pronormal.

Related facts

Related facts about pronormality

• Pronormality is normalizing join-closed
• Pronormality is not finite-join-closed
• Pronormality is not commutator-closed
• Pronormality is not centralizer-closed

Related facts about permuting joins

• Subnormality is permuting join-closed
• Subnormality is normalizing join-closed

Proof

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