# Center not is intermediately powering-invariant

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
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

## 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 $H$.

## Proof

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.