Shintani descent, simple groups and spread

TL;DR

This paper investigates the spread of almost simple groups, using Shintani descent to classify when the spread tends to infinity in sequences of such groups, and explores properties of maximal overgroups.

Contribution

It introduces a self-contained approach to Shintani descent, applying it to analyze the spread and maximal overgroups in almost simple groups, extending previous classifications.

Findings

Characterizes when the spread of almost simple groups diverges

Shows Shintani descent preserves maximal overgroup information

Establishes bounds on the minimal number of maximal overgroups

Abstract

The spread of a group $G$, written $s(G)$, is the largest $k$ such that for any nontrivial elements $x_1, \dots, x_k \in G$ there exists $y \in G$ such that $G = \langle x_i, y \rangle$ for all $i$. Burness, Guralnick and Harper recently classified the finite groups $G$ such that $s(G) > 0$, which involved a reduction to almost simple groups. In this paper, we prove an asymptotic result that determines exactly when $s(G_n) \to \infty$ for a sequence of almost simple groups $(G_n)$. We apply probabilistic and geometric ideas, but the key tool is Shintani descent, a technique from the theory of algebraic groups that provides a bijection, the Shintani map, between conjugacy classes of almost simple groups. We provide a self-contained presentation of a general version of Shintani descent, and we prove that the Shintani map preserves information about maximal overgroups. This is suited to further applications. Indeed, we also use it to study $\mu(G)$, the minimal number of maximal overgroups of an element of $G$. We show that if $G$ is almost simple, then $\mu(G) \leqslant 3$ when $G$ has an alternating or sporadic socle, but in general, unlike when $G$ is simple, $\mu(G)$ can be arbitrarily large.