The minimal size of a generating set for primitive $\frac{3}{2}$-transitive groups

TL;DR

This paper establishes that all primitive 3/2-transitive groups are generated by at most four elements, with most being generated by two, and provides a complete description of exceptions, contributing to the understanding of minimal generating sets.

Contribution

It proves that primitive 3/2-transitive groups have a minimal generating set of size at most four, and most are two-generated, offering a complete classification of certain solvable affine groups.

Findings

All primitive 3/2-transitive groups are at most 4-generated.

Most such groups are 2-generated, except specific solvable affine groups.

Groups with cyclic abelian subgroups and Frobenius complements are 2-generated.

Abstract

We refer to $d(G)$ as the minimal cardinality of a generating set of a finite group $G$, and say that $G$ is $d$-generated if $d(G)\leq d$. A transitive permutation group $G$ is called $\frac{3}{2}$-transitive if a point stabilizer $G_\alpha$ is nontrivial and its orbits distinct from $\{\alpha\}$ are of the same size. We prove that $d(G)\leq4$ for every primitive $\frac{3}{2}$-transitive permutation group $G$, moreover, $G$ is $2$-generated except for the very particular solvable affine groups that we completely describe. In particular, all finite $2$-transitive and $2$-homogeneous groups are $2$-generated. We also show that every finite group whose abelian subgroups are cyclic is $2$-generated, and so is every Frobenius complement.