# Omega-1 of center not is minimal 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., Omega-1 of center) doesnotalways satisfy a particular subgroup property (i.e., minimal characteristic subgroup)

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

## Statement

Let be a prime number and be a nilpotent p-group. Then, is *not necessarily* a minimal characteristic subgroup.

## Proof

### Example of an Abelian group

Consider . Then, . On the other hand, the subgroup is a *strictly smaller* nontrivial characteristic subgroup of .