On the minimal degree and base size of finite primitive groups

Let G be a finite permutation group acting on Omega. A base for G is a subset B subset of Omega such that the pointwise stabilizer G(B) is the identity. The base size of G, denoted by b(G), is the cardinality of the smallest possible base. The minimal degree of G, denoted by mu(G), is the smallest cardinality of the support of a non-trivial element of G. In this paper, we establish a new upper bound for b(G) when G is primitive, and subsequently prove that if G is a primitive group different from the Mathieu group of degree 24, then mu(G)b(G) <= nlog n, where n is the degree of G. This bound is best possible, up to a multiplicative constant.