Powering-invariance is not finite-join-closed

Statement
It is possible to have a group $$G$$ and subgroups $$H$$ and $$K$$ of $$G$$ such that both $$H$$ and $$K$$ are both powering-invariant subgroups of $$G$$ but the join of subgroups $$\langle H, K \rangle$$ is not a powering-invariant subgroup of $$G$$.

Related facts

 * Powering-invariance is strongly intersection-closed
 * Divisibility-closedness is not finite-join-closed
 * Divisibility-closedness is not finite-intersection-closed

Nilpotent case

 * Powering-invariance is strongly join-closed in nilpotent group
 * Divisibility-closedness is strongly join-closed in nilpotent group

Proof
Suppose $$G$$ is the generalized dihedral group corresponding to the additive group of rational numbers. Let $$H$$ and $$K$$ both be subgroups of order two generated by different reflections. Then, the following are true:


 * $$G$$ is powered over all primes other than 2.
 * $$H$$ and $$K$$ are both powering-invariant subgroups on account of being finite subgroups (see finite implies powering-invariant).
 * $$\langle H, K \rangle$$ is isomorphic to the infinite dihedral group. It is not powered over any primes, and in particular it is not powering-invariant in $$G$$.