Minimal degree of primitive permutation groups

Abstract

If G is a permutation group on a set Ω of n points then the minimal number c of points of Ω permuted by non-identity elements of G is called the minimal degree of G. If G is primitive then Jordan (1871) showed that as n gets large so does c. Later in 1892 and 1897, Bochert obtained a simple bound for c in terms of n provided that G is 2-transitive and is not the alternating or symmetric group: he showed that c ⩾ n/4−1 (in 1892), and c ⩾ n/3−2√n/3 (in 1897). This paper is the result of our efforts to obtain simpler bounds than those of Jordan when G is primitive but not 2-transitive. We show that if G is primitive on Ω of rank r ⩾ 3 and minimal degree c, and if n_min is the minimal length of the orbits of G_α in Ω-{α}, where α ε Ω, then c⩾n_min/4+r−1. Moreover as two corollaries of the result we show that if either G has rank 3, or if G is 3/2-transitive then c is of the order of √n, where n=|Ω|, which is better than the bounds of Jordan.