Maximal $\mathrm{PS}{\mathrm{L}}_2$ subgroups of exceptional groups of Lie type

Abstract

We study embeddings of ${\mathrm{PSL}}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2{E}_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle ${\mathrm{PSL}}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2{E}_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ 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 ${\mathrm{PSL}}_2(p^a)$ inside these finite almost simple groups are known, with three possible exceptions ($p^a=7,8,25$ for $E_7$). In the three remaining cases we provide considerable information about a potential maximal subgroup.

v+154pp. Mem. Amer. Math. Soc., to appear

References in corpus (2)

• Alternating Subgroups of Exceptional Groups of Lie Type
• Subspace stabilizers and maximal subgroups of exceptional groups of Lie type