Asymptotic Results for Transitive Permutation Groups

Abstract

In this paper we give answers to some open questions concerning generation and enumeration of finite transitive permutation groups. In [1], Bryant, Kovács and Robinson proved that there is a number c′ such that each soluble transitive permutation group of degree n⩾2 can be generated by [ c ′ n / l o g n ] elements, and later A. Lucchini [5] extended this result (with a different constant c′) to finite permutation groups containing a soluble transitive subgroup. We are now able to prove this theorem in full generality, and this solves the question of bounding the number of generators of a finite transitive permutation group in terms of its degree. The result obtained is the following. 1991 Mathematics Subject Classification 20B05, 20D60.