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