# Powering-invariant and normal not implies quotient-powering-invariant

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., powering-invariant normal subgroup) neednotsatisfy the second subgroup property (i.e., quotient-powering-invariant subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about powering-invariant normal subgroup|Get more facts about quotient-powering-invariant subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property powering-invariant normal subgroup but not quotient-powering-invariant subgroup|View examples of subgroups satisfying property powering-invariant normal subgroup and quotient-powering-invariant subgroup

## Statement

It is possible to have a group and a powering-invariant normal subgroup of that is not quotient-powering-invariant: in other words, is a normal subgroup of that is also a powering-invariant subgroup of , but there exists a prime number such that is -powered and is not -powered.

We can in fact construct our example such that both and are rationally powered groups and contains a divisible group as subgroup where every element has infinitely many roots for all .