The classification of (3/2)-transitive permutation groups and (1/2)-transitive linear groups

TL;DR

This paper classifies all finite (1/2)-transitive linear groups and, as a result, completes the classification of finite (3/2)-transitive permutation groups, also extending to (k+1/2)-transitive groups for k > 1.

Contribution

It provides a complete classification of (1/2)-transitive linear groups and (3/2)-transitive permutation groups, extending to higher half-integer transitivity levels.

Findings

Complete classification of (1/2)-transitive linear groups.

Complete classification of (3/2)-transitive permutation groups.

Extension to (k+1/2)-transitive groups for k > 1.

Abstract

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear groups. As a consequence we complete the determination of the finite (3/2)-transitive permutation groups -- the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the finite (k+1/2)-transitive permutation groups for integers k > 1.