Let V V be a finite vector space over a finite field of order q q and of characteristic p p . Let G ≤ G L ( V ) G\leq GL(V) be a p p -solvable completely reducible linear group. Then there exists a base for G G on V V of size at most 2 2 unless q ≤ 4 q \leq 4 in which case there exists a base of size at most 3 3 . This extends a recent result of Halasi and Podoski and generalizes a theorem of Seress. A generalization of a theorem of Pálfy and Wolf is also given.