Maximal PSL₂ subgroups of exceptional groups of Lie type

2016/10/24 by Craven, David A.
#20D06 #20E28 #20G40 #20G41 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1610.07469

Abstract

We study embeddings of PSL₂(p^a) into exceptional groups G(p^b) for G=F₄,E₆,² E₆,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, ² E₆ 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.

Related

• Embeddings of PSL₂(q) in exceptional groups of Lie type over a field of characteristic ≠2,3
• Maximal subgroups of almost simple groups with socle \PSL(2,q)
• On the Maximal Subgroups of E₇(q) and Related Almost Simple Groups
• Exceptional Lie groups
• Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic