Shintani descent, simple groups and spread

JOURNAL OF ALGEBRA

卷:578

Article

Harper, Scott^1,2 

[1] Univ Bristol, Sch Math, Bristol BS8 1UG, Avon, England

[2] Heilbronn Inst Math Res, Bristol, Avon, England

关键词: Almost simple groups;    Maximal subgroups;    Shintani descent;    Spread;   

DOI  :  10.1016/j.jalgebra.2021.02.021

来源: Elsevier

【 摘 要 】

The spread of a group G, written s(G), is the largest k such that for any nontrivial elements x(1), ..., x(k) is an element of G there exists y is an element of 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)) -> infinity 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) <= 3 when G has an alternating or sporadic socle, but in general, unlike when G is simple, mu(G) can be arbitrarily large. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free