# Powering-invariance is not finite-join-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup)notsatisfying a subgroup metaproperty (i.e., finite-join-closed group property).

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## Statement

It is possible to have a group and subgroups and of such that both and are both powering-invariant subgroups of but the join of subgroups is not a powering-invariant subgroup of .

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

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