The minimal base size for a p-solvable linear group

Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $Gleq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Palfy and Wolf is also given.