Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set Ω. A sequence B =(ω 1 , …, ω b ) of points in Ω is called a base if its pointwise stabilizer in G is the identity. Bases are of fundamental importance in computational algorithms for permutation groups. For both practical and theoretical reasons, one is interested in the minimal base size for ( G , Ω), For a nonredundant base B , the elementary inequality 2 ∣ B ∣ ⩽∣ G ∣⩽∣Ω∣ ∣ B ∣ holds; in particular, ∣ B ∣⩾log∣ G ∣/log∣Ω∣. In the case when G is primitive on Ω, Pyber [ 8 , p. 207] has conjectured that the minimal base size is less than C log∣ G ∣/log∣Ω∣ for some (large) universal constant C . It appears that the hardest case of Pyber's conjecture is that of primitive affine groups. Let H = GV be a primitive affine group; here the point stabilizer G acts faithfully and irreducibly on the elementary abelian regular normal subgroup V of H , and we may assume that Ω= V . For positive integers m , let mV denote the direct sum of m copies of V . If ( v 1 , …, v m )∈ mV belongs to a regular G ‐orbit, then (0, v 1 , …, v m ) is a base for the primitive affine group H . Conversely, a base (ω 1 , …, ω b ) for H which contains 0∈ V =Ω gives rise to a regular G ‐orbit on ( b −1) V . Thus Pyber's conjecture for affine groups can be viewed as a regular orbit problem for G ‐modules, and it is therefore a special case of an important problem in group representation theory. For a related result on regular orbits for quasisimple groups, see [ 4 , Theorem 6].