On minimal degrees and base sizes of primitive permutation groups

Abstract

The minimal degree/~ (G) of a primitive permutat ion group G of degree n on a set ~, that is, the smallest number of points moved by any non-identity element of G, has been the subject of considerable study in the classical theory of permutat ion groups (see w 15 of [9], for example). The best result available before the completion of the classification of finite simple groups was one due to Babai (see Theorem 6.14 of [1]): provided that A, :g G, we have #(G) > 89 1). In this note we use the classification theorem to improve this result. This is done by consideration of bases of G: a subset A of O is a base of G if the pointwise stabiliser of A in G is the identity (A is called afixing set in [1]). Denote by b (G) the minimum size of any base of G. It is shown in [1] that if G is simply primitive then b (G) < 4 x /~ log n. Using the classification theorem we prove the following result.