Hi: In the following, G is a group that acts on the set . For .

Proposition: Suppose that G contains a normal subgroup N, which acts transitively on . Then for every . In particular, is a complement of N in G if .

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 and . The transitivity of on gives an element such that . Hence and thus . This shows that .