Shintani descent, simple groups and spread

DOI10.1016/J.JALGEBRA.2021.02.021arXiv2008.02558MaRDI QIDQ6346598

Scott Harper

Publication date: 6 August 2020

Abstract: The spread of a group $G$, written $s(G)$, is the largest $k$ such that for any nontrivial elements $x_1,dots,x_{k}inG$ there exists $yinG$ such that $G=langle{x}_i,yangle$ 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 $s(G_n)oinfty$ 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 $mu(G)leqslant3$ when $G$ has an alternating or sporadic socle, but in general, unlike when $G$ is simple, $mu(G)$ can be arbitrarily large.

Mathematics Subject Classification ID

Subgroup theorems; subgroup growth (20E07) Maximal subgroups (20E28) Generators, relations, and presentations of groups (20F05)

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