Large orbits in coprime actions of solvable groups.

Let $G$ be a solvable group of automorphisms of a finite group $K$. If $\abs{G}$ and $\abs{K}$ are coprime, then there exists an orbit of $G$ on $K$ of size at least $\sqrt{\abs{G}}$. It is also proved that in a $\pi$-solvable group, the largest normal $\pi$-subgroup is the intersection of at most three Hall $\pi$-subgroups.