# Complements to normal subgroup need not be automorphic

## Contents

## Statement

Suppose is a group, is a normal subgroup, and and are permutable complements to in . Then, it is *not* necessary that there exists an automorphism of sending to .

## Related facts

## Proof

### A generic example

Let be any non-Abelian group. Consider and the subgroup . Let be the subgroup and be the subgroup .

Note that:

- is normal in : In fact, it is a direct factor of .
- is a permutable complement to in .
- is a permutable complement to in .
- is normal in : In fact, it is a direct factor of .
- is not normal in : Pick such that do not commute. Then, we have . Thus, a conjugate of an element in lies outside .