# Center not is intermediately characteristic

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 characteristic subgroup)

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

## Statement

The center of a group need not be characteristic in every intermediate subgroup.

## Proof

### Example of the dihedral group

In the dihedral group of order eight, generated by a rotation and reflection , the center is *not* intermediately characteristic. In fact, it is not characteristic in the intermediate subgroup .