Permutability is not upper join-closed

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

It is possible to have a group $G$, a subgroup $H$, and intermediate subgroups $K_{1},K_{2}$ of $G$ containing $H$ such that $H$ is a permutable subgroup in both $K_{1}$ and $K_{2}$, but $H$ is not a permutable subgroup in $\langle K_{1},K_{2}\rangle$.

Related facts

Related facts that don't hold for permutable subgroups

• Permutable not implies normal
• Permutability is not finite-intersection-closed: The same example setup works as the example setup for the proof of this statement.

Related facts that do hold for permutable subgroups

• Permutability is strongly join-closed
• Permutability satisfies intermediate subgroup condition
• Permutability satisfies image condition
• Permutability satisfies inverse image condition
• Permutability satisfies transfer condition

Proof

Construction of the counterexample

Setup: Let $p$ be an odd prime.

• $A$ is a group generated by two elements $a,b$ subject to the relations $a^{p^{2}}=1,b^{p}=1$ and $ab=ba^{p+1}$. Alternatively $A$ is the semidirect product of the additive group modulo $p^{2}$ by the multiplicative group of order $p$ in the multiplicative group of automorphisms. Note that $A$ is a non-Abelian group of order $p^{3}$.
• $C$ is a cyclic group of order $p^{2}$, generated by an element $c$.
• $G=A\times C$.
• $B=\{b\}$.
• $K_{1}=A\times \{e\}$, and $K_{2}=B\times C=\{b,c\}$.
• $H=B\times \{e\}$.

We claim that $H$ is permutable in both $K_{1}$ and $K_{2}$ but not in $G$.

• $H$ is permutable in $K_{1}$: This follows from the fact that $B$ is permutable in $A$. This can be verified using the fact that it is in the Baer norm, or equivalently, that it commutes with all cyclic subgroups. The proof details are given in the example for permutable not implies normal.
• $H$ is permutable in $K_{2}$: This follows from the fact that $H$ is a direct factor of $K_{2}$, hence normal in $K_{2}$, hence permutable in $K_{2}$.
• $H$ is not permutable in $G$: Consider the cyclic subgroup $D$ generated by $(a,c)$. The claim is that $HD\neq DH$. To prove this notice that $DH\ni (a,c)(b,e)=(ab,c)=(ba^{p+1},c)$. This is clearly not in $HD$.