On Strongly P-embedded Subgroups of Lie Rank 2

For a prime p, a proper subgroup H of the finite group G is strongly p-embedded in G if p divides |H| and p does not divide |H ∩ H| for all g ∈ G \H . A characteristic property of strongly p-embedded subgroups is that NG(X) ≤ H for any non-trivial p-subgroup X of H . Strongly p-embedded subgroups appear in the final stages of one of the programmes to better understand the classification of the finite simple groups [7]. The almost simple groups with strongly 2-embedded subgroups were determined by Bender [2] and Suzuki [10]. Recall that a K-group is a group in which every composition factor is from the list of “known” simple groups. That is, every simple section is either a cyclic group of prime order, an alternating group, a group of Lie type or one of the twenty six sporadic simple groups. In the classification of groups of local characteristic p, certain normalizers of non-trivial subgroups are assumed to be K-groups. The objective of this paper is to extend the results of [9, Corollary 1.4] to cover some of the Lie type groups of Lie rank 2 defined in characteristic p. Further results related to local characteristic p identifications of rank 2 Lie type groups can be found in the work of Parker and Rowley [8]. Our main result in this article is the following theorem.