# Powering-invariance does not satisfy intermediate subgroup condition

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., intermediate subgroup condition).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about powering-invariant subgroup|Get more facts about intermediate subgroup condition|

## Statement

It is possible to have groups such that is a powering-invariant subgroup of but not of .

## Proof

Take:

- to be the group .
- to be the subgroup .
- to be the subgroup .

Then:

- is powering-invariant in : is not powered over any primes, so is by definition powering-invariant in .
- is not powering-invariant in : is powered over all primes, and is not powered over any, so is not powering-invariant in .