# Difference between revisions of "Powering-invariance is not finite-join-closed"

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) not satisfying a subgroup metaproperty (i.e., finite-join-closed group property).
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## 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 $G$.

## 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 $G$.