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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)
View subgroup property satisfactions for subgroup-defining functions | View subgroup property dissatisfactions for subgroup-defining functions

Contents

Statement

It is possible to have a group G such that the center Z(G) is not an intermediately powering-invariant subgroup of G, i.e., there exists an intermediate subgroup H of G such that Z(G) is not a powering-invariant subgroup of H.

Related facts

Proof

Further information: Amalgamated free product of two copies of group of rational numbers over group of integers

Let G be the group \mathbb{Q} *_{\mathbb{Z}} \mathbb{Q} and H be the first factor \mathbb{Q} as a subgroup of G. Then, Z(G) inside H is like Z in Q, and it is not powering-invariant.