Base sizes of primitive groups: bounds with explicit constants

Abstract

We show that the minimal base size b ( G )of a finite primitive permutation group G of degree n is at most 2(log | G | / log n ) + 24. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups G of degrees n such that b ( G ) = 2(log | G | / log n ) − 2 and b ( G )is unbounded. As a corollary we show that a primitive permutation group of degree n that does not contain the alternating group Alt( n )has a base of size at most max { √ n, 25 }