Let $K$ be a normal subgroup of $G$ and let $G$ act on a set $X$. Suppose the action of $K$ on $X$ is free and transitive. Then the stabilizer of any element in $X$ forms a complement of $K$.
I was wondering if anyone knows of a reference for this (or an equivalent of this) statement. I have only been able to find this stackexchange post (Group Theory. Transitivity and Normal subgroups.).
