# Center not is intermediately powering-invariant

From Groupprops

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) doesnotalways 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 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 .

## Related facts

### Opposite facts

## Proof

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

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.