Shintani descent, simple groups and spread

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 G=〈xi,y〉 all i. Burness, Guralnick and Harper recently classified finite groups G s(G)>0, which involved reduction to almost simple groups. In this paper, we prove an asymptotic result determines exactly when s(Gn)→∞ sequence (Gn). We apply probabilistic geometric ideas, but key tool Shintani descent, technique from theory algebraic provides bijection, map, between conjugacy classes provide self-contained presentation general version map preserves information about maximal overgroups. This suited further applications. Indeed, also use it study μ(G), minimal number overgroups element G. show if simple, then μ(G)⩽3 has alternating or sporadic socle, in general, unlike μ(G) can be arbitrarily large.