  Let $V$ be a finite vector space over a finite field of order $q$ and of
characteristic $p$. Let $G\leq GL(V)$ 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 P\'alfy
and Wolf is also given.
