Retract not implies every permutable complement is normal

Statement

Suppose $G$ is a group and $H$ is a retract of $G$. In other words, $H$ has a normal complement in $G$: there exists a normal subgroup $N$ of $G$ such that $N\cap H$ is trivial and $NH=G$.

There may exist non-normal permutable complements to $H$ in $G$. In other words, there may exist a non-normal subgroup $K$ of $G$ such that $HK=G$ and $H\cap K$ 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 $G$ be the symmetric group on the set $\{1,2,3,4\}$, $H$ be the subgroup comprising permutations on $\{1,2,3\}$. Then:

• $H$ has a normal complement, namely the subgroup $N=\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$.
• $H$ has a permutable complement that is not normal, namely the subgroup $K=\{(),(1,2,3,4),(1,3)(2,4),(1,4,3,2)\}$.