Shintani descent, simple groups and spread

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 $G=⟨x_i,y⟩$ 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 $\hat{I}¼(G)$, the minimal number of maximal overgroups of an element of $G$. We show that if $G$ is almost simple, then $\hat{I}¼(G)⩽3$ when $G$ has an alternating or sporadic socle, but in general, unlike when $G$ is simple, $\hat{I}¼(G)$ can be arbitrarily large.

30 pages; to appear in Journal of Algebra