Bounds for Finite Semiprimitive Permutation Groups: Order, Base Size, and Minimal Degree

TL;DR

This paper establishes bounds on key parameters of finite semiprimitive permutation groups, such as order and base size, by classifying groups with specific induced actions, advancing understanding of their structure.

Contribution

It provides new bounds for order, base size, and minimal degree of finite semiprimitive groups and classifies those inducing symmetric or alternating groups on orbit sets.

Findings

Bounds on order, base size, and minimal degree in terms of degree

Classification of groups inducing symmetric or alternating groups

Enhanced understanding of semiprimitive group structure

Abstract

In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.