Center not is intermediately powering-invariant
This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., intermediately powering-invariant subgroup)
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It is possible to have a group such that the center is not an intermediately powering-invariant subgroup of , i.e., there exists an intermediate subgroup of such that is not a powering-invariant subgroup of .
Let be the group and be the first factor as a subgroup of . Then, inside is like Z in Q, and it is not powering-invariant.