Maximal PSL2 subgroups of exceptional groups of Lie type

David Craven

doi:10.1090/memo/1355

Maximal PSL2 subgroups of exceptional groups of Lie type

David Craven

Mathematics

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Abstract

We study embeddings of PSL₂(p^a) into exceptional groups G(p^b) for G = F₄, E₆, ²E₆, E₇, and p a prime with a, b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL₂(p^a), that is maximal inside an almost simple exceptional group of Lie type F₄, E₆, ²E₆ and E₇, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A₁ inside the algebraic group.

Together with a recent result of Burness and Testerman for p the Coxeter number plus one, this proves that all maximal subgroups with socle PSL₂(p^a) inside these finite almost simple groups are known, with three possible exceptions (p^a = 7, 8, 25 for E₇).

In the three remaining cases we provide considerable information about a potential maximal subgroup.

Original language	English
Pages (from-to)	1-168
Journal	Memoirs of the American Mathematical Society
Volume	276
Issue number	1355
Early online date	25 Feb 2022
DOIs	https://doi.org/10.1090/memo/1355
Publication status	Published - Mar 2022

Keywords

Maximal subgroups
exceptional groups
finite simple groups

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10.1090/memo/1355

CravenD2019MaximalAccepted author manuscript, 1.2 MBLicence: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)

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title = "Maximal PSL2 subgroups of exceptional groups of Lie type",

abstract = "We study embeddings of PSL2(pa) into exceptional groups G(pb) for G = F4, E6, 2E6, E7, and p a prime with a, b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(pa), that is maximal inside an almost simple exceptional group of Lie type F4, E6, 2E6 and E7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1 inside the algebraic group. Together with a recent result of Burness and Testerman for p the Coxeter number plus one, this proves that all maximal subgroups with socle PSL2(pa) inside these finite almost simple groups are known, with three possible exceptions (pa = 7, 8, 25 for E7). In the three remaining cases we provide considerable information about a potential maximal subgroup. ",

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AU - Craven, David

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N2 - We study embeddings of PSL2(pa) into exceptional groups G(pb) for G = F4, E6, 2E6, E7, and p a prime with a, b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(pa), that is maximal inside an almost simple exceptional group of Lie type F4, E6, 2E6 and E7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1 inside the algebraic group. Together with a recent result of Burness and Testerman for p the Coxeter number plus one, this proves that all maximal subgroups with socle PSL2(pa) inside these finite almost simple groups are known, with three possible exceptions (pa = 7, 8, 25 for E7). In the three remaining cases we provide considerable information about a potential maximal subgroup.

AB - We study embeddings of PSL2(pa) into exceptional groups G(pb) for G = F4, E6, 2E6, E7, and p a prime with a, b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(pa), that is maximal inside an almost simple exceptional group of Lie type F4, E6, 2E6 and E7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1 inside the algebraic group. Together with a recent result of Burness and Testerman for p the Coxeter number plus one, this proves that all maximal subgroups with socle PSL2(pa) inside these finite almost simple groups are known, with three possible exceptions (pa = 7, 8, 25 for E7). In the three remaining cases we provide considerable information about a potential maximal subgroup.

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KW - exceptional groups

KW - finite simple groups

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EP - 168

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

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Maximal PSL2 subgroups of exceptional groups of Lie type

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