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

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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) need not satisfy the second subgroup property (i.e., quotient-powering-invariant subgroup)
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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 G and a powering-invariant normal subgroup H of G that is not quotient-powering-invariant: in other words, H is a normal subgroup of G that is also a powering-invariant subgroup of G, but there exists a prime number p such that G is p-powered and G/H is not p-powered.

We can in fact construct our example such that both G and H are rationally powered groups and G/H contains a divisible group as subgroup where every element has infinitely many n^{th} roots for all n > 1.

References