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 = hxi, yi 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) 6 3 when G has an alternating or sporadic socle, but in general, unlike when G is simple, µ(G) can be arbitrarily large.
Bibliographical noteFunding Information:
The author is grateful to the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Groups, Representations and Applications: New perspectives, when some work on this paper was undertaken. The author thanks Jean Michel for helpful comments he made on the author's talk during that programme. The author also thanks Tim Burness, Gunter Malle and an anonymous referee for useful feedback on previous versions of this paper. This work was supported by EPSRC grant number EP/R014604/1.
© 2021 Elsevier Inc.
- Almost simple groups
- Maximal subgroups
- Shintani descent