Hi: In the following, G is a group that acts on the set \Omega. For \alpha \in \Omega, G_\alpha := \{x \in G |  \alpha x = \alpha\}.

Proposition: Suppose that G contains a normal subgroup N, which acts transitively on \Omega. Then G = G_\alpha N for every \alpha \in \Omega. In particular, G_\alpha is a complement of N in G if N_\alpha= 1.

This proposition is in a book, Kurzweil - Stellmacher, The Theory of Finite Groups, An Introduction, Springer, 2004. What I do not understand is why N has to be normal. In the proof given by the author, no use is made of that fact. The proof runs like this:

Let \alpha \in \Omega and y \in G. The transitivity of N on \Omega gives an element x \in N such that \alpha y = \alpha x. Hence \alpha yx^{-1} = \alpha and thus yx^{-1} \in G_\alpha. This shows that y \in G_\alpha x \subseteq G_\alpha N.