Maximal 𝑃𝑆𝐿₂ Subgroups of Exceptional Groups of Lie Type

Abstract

We study embeddings of P S L 2 ( p a ) \mathrm {PSL}_2(p^a) into exceptional groups G ( p b ) G(p^b) for G = F 4 , E 6 , 2 E 6 , E 7 G=F_4,E_6,{}^2\!E_6,E_7 , and p p a prime with a , b a,b positive integers. With a few possible exceptions, we prove that any almost simple group with socle P S L 2 ( p a ) \mathrm {PSL}_2(p^a) , that is maximal inside an almost simple exceptional group of Lie type F 4 F_4 , E 6 E_6 , 2 E 6 {}^2\!E_6 and E 7 E_7 , is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A 1 A_1 inside the algebraic group. Together with a recent result of Burness and Testerman for p p the Coxeter number plus one, this proves that all maximal subgroups with socle P S L 2 ( p a ) \mathrm {PSL}_2(p^a) inside these finite almost simple groups are known, with three possible exceptions ( p a = 7 , 8 , 25 p^a=7,8,25 for E 7 E_7 ). In the three remaining cases we provide considerable information about a potential maximal subgroup.