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 {