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 x1,…,xk∈G there exists y∈G such that G=〈xi,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(Gn)→∞ for a sequence of almost simple groups (Gn). 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 μ(G), the minimal number of maximal overgroups of an element of G. We show that if G is almost simple, then μ(G)⩽3 when G has an alternating or sporadic socle, but in general, unlike when G is simple, μ(G) can be arbitrarily large.