The classification of \frac{3}2-transitive permutation groups and \frac{1}2-transitive linear groups

Abstract

A linear group G ≤ G L ( V ) G\le GL(V) , where V V is a finite vector space, is called 1 2 \frac {1}{2} -transitive if all the G G -orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1 2 \frac {1}{2} -transitive linear groups. As a consequence we complete the determination of the finite 3 2 \frac {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 ( k + 1 2 ) (k+\frac {1}{2}) -transitive groups for integers k ≥ 2 k\ge 2 .