# Center not is endomorphism kernel

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., endomorphism kernel)

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 endomorphism kernel in , i.e., does not have any subgroup isomorphic to its inner automorphism group.

## Proof

`Further information: subgroup structure of quaternion group`

The simplest example is where is the quaternion group, so is the center of quaternion group and the quotient group is isomorphic to the Klein four-group. has no subgroup isomorphic to the Klein four-group.