Frobenius Theorem

Definition

A Frobenius group is a group $G$ with a faithful transitive action on a set $X$ such that no non-trivial element fixes more than one point.

Theorem

For a Frobenius group $G$, its regular elements and $1$ form a characteristic subgroup of $G$, which is called Frobinuous complement.

Remark. By Frattini’s argument, for a Frobenius kernel $K$, it has a complement $H=G_{\omega }$ such that $G=K:G_{\omega }$ for any $\omega \in X$.