# Center not is surjective endomorphism-balanced

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., surjective endomorphism-balanced subgroup)

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

## Contents

## Statement

### Statement with symbols

It is possible to have a group and a surjective endomorphism of such that the restriction of to the center is *not* surjective as an endomorphism of .

## Related facts

### Other proofs of similar subgroup-defining functions not being surjective endomorphism-balanced

- Baer norm not is surjective endomorphism-balanced
- Wielandt subgroup not is surjective endomorphism-balanced

### Proofs of similar subgroup-defining functions being strictly characteristic

- Center is strictly characteristic
- Baer norm is strictly characteristic
- Wielandt subgroup is strictly characteristic

## Facts used

- Isomorphic to inner automorphism group not implies centerless: We can have a group such that the center is nontrivial, but there is an isomorphism between and .

## Proof

### Proof using fact (1)

Let be an example group for fact (1). Let and the quotient map and be an isomorphism. Then, is a surjective endomorphism of . On the other hand, the restriction of to is the trivial map, which is *not* surjective by the assumption that is a nontrivial group.