# Retract not implies every permutable complement is normal

From Groupprops

## Contents

## Statement

Suppose is a group and is a retract of . In other words, has a normal complement in : there exists a normal subgroup of such that is trivial and .

There may exist *non*-normal permutable complements to in . In other words, there may exist a non-normal subgroup of such that and is trivial.

## Related facts

- Retract not implies normal complements are isomorphic
- Every group of given order is a permutable complement for symmetric groups

## Proof

### Example

`Further information: symmetric group:S4`

Let be the symmetric group on the set , be the subgroup comprising permutations on . Then:

- has a normal complement, namely the subgroup .
- has a permutable complement that is not normal, namely the subgroup .