# Powering-invariance is not commutator-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., commutator-closed subgroup property).

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## Statement

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

## Related facts

- Powering-invariance is commutator-closed in nilpotent group
- Powering-invariance is not finite-join-closed
- Powering-invariance 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 an infinite cyclic group, isomorphic to the group of integers. It is not powered over any prime, hence is not powering-invariant in .