On Pyber’s base size conjecture

Abstract

Abstract. Let G be a permutation group on a finite set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. The base size of G, denoted b(G), is the smallest size of a base. A well known conjecture of Pyber from the early 1990s asserts that there exists an absolute constant c such that b(G) c log |G|/ log n for any primitive permutation group G of degree n. Some special cases have been verified in recent years, including the almost simple and diagonal cases. In this paper, we prove Pyber's conjecture for all non-affine primitive groups.

Base sizes of primitive groups: Bounds with explicit constants

“…Assume that H ≤ Sym(Γ) is an almost simple group with socle T . We follow not only [12] here but [13, §4]. However we avoid the use of the bound |Out(T )| ≤ |T | α , since this is expensive.…”

A proof of Pyber's base size conjecture

“…(Moreover, they form a system of blocks for the action of G on Γ.) Let us denote the size of each part I j by t. In accordance with [14], we will refer to this number as the linking factor of N . Thus, we have…”

“…Thus we may assume that H is of diagonal type. In this case, by an argument different from the one in [14,Proposition 3.10], it is possible to obtain a bound with multiplicative constant less than 22. For the details, see [24,Section 4.3.2] by Liebeck and the second and third authors.…”

“…Primitive permutation groups of diagonal type were handled by Gluck, Seress, Shalev [21,Remark 4.3] and Fawcett [18]. For primitive groups of product type and of twisted wreath product type the conjecture was established by Burness and Seress [14]. From these results one can deduce the general bound b(G) < 45 log |G| log n for a non-affine primitive permutation group G of degree n.…”

“…Clearly, if G is a permutation group of degree n > 1, then n |G| < d(G). Burness and Seress [14] stated (with a different languague) that there exists a universal constant c > 0 such that d(G) ≤ |G| c/n provided that G is a transitive permutation group of degree n (see also Theorem 2.2 and the discussion preceding it). The proof of this latter fact misses a case.…”