Shintani descent, simple groups and spread
Date
15/07/2021Author
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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.
Citation
Harper , S 2021 , ' Shintani descent, simple groups and spread ' , Journal of Algebra , vol. 578 , pp. 319-355 . https://doi.org/10.1016/j.jalgebra.2021.02.021
Publication
Journal of Algebra
Status
Peer reviewed
ISSN
0021-8693Type
Journal article
Rights
Copyright © 2021 Elsevier Inc. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jalgebra.2021.02.021.
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