On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

Abstract

We study embeddings of groups of Lie type H H in characteristic p p into exceptional algebraic groups G \mathbf {G} of the same characteristic. We exclude the case where H H is of type P S L 2 \mathrm {PSL}_2 . A subgroup of G \mathbf {G} is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of G \mathbf {G} . With a few possible exceptions, we prove that there are no Lie primitive subgroups H H in G \mathbf {G} , with the conditions on H H and G \mathbf {G} given above. The exceptions are for H H one of P S L 3 ( 3 ) \mathrm {PSL}_3(3) , P S U 3 ( 3 ) \mathrm {PSU}_3(3) , P S L 3 ( 4 ) \mathrm {PSL}_3(4) , P S U 3 ( 4 ) \mathrm {PSU}_3(4) , P S U 3 ( 8 ) \mathrm {PSU}_3(8) , P S U 4 ( 2 ) \mathrm {PSU}_4(2) , P S p 4 ( 2 ) ′ \mathrm {PSp}_4(2)’ and 2 B 2 ( 8 ) {}^2\!B_2(8) , and G \mathbf {G} of type E 8 E_8 . No examples are known of such Lie primitive embeddings. We prove a slightly stronger result, including stability under automorphisms of G \mathbf {G} . This has the consequence that, with the same exceptions, any almost simple group with socle H H , 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 , E 7 E_7 and E 8 E_8 , is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group. The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.