Strongly P-embedded Subgroups

In this paper we study finite groups which possess a strongly pembedded subgroup for some prime p. Suppose that p is a prime. A subgroup H of the finite group G is said to be strongly p-embedded in G if the following two conditions hold. (i) H < G and p divides |H|; and (ii) if g ∈ G \H , then p does not divide |H ∩H|. One of the most important properties of strongly p-embedded subgroups is that NG(X) ≤ H for any non-trivial p-subgroup X of H . Groups with a strongly 2-embedded subgroup have been classified by Bender [2] and Suzuki [15] and their classification forms a pedestal upon which the classification of the finite simple groups stands. The classification of groups with a strongly 2-embedded subgroup states that if G is a finite group with a strongly 2-embedded subgroup, then O ′ (G/O(G)) is a simple rank 1 Lie type group defined in characteristic 2 or G has quaternion or cyclic Sylow 2-subgroups. Of course rank 1 Lie type groups in characteristic 2 are the building blocks of the groups of Lie type in characteristic 2. When dealing with groups defined in characteristic p for odd p, strongly p-embedded subgroups play an equally influential role. Assume from here on that p is an odd prime and G is a finite group. If G has cyclic Sylow p-subgroup P , then NG(Ω1(P )) is strongly pembedded in H . There is thus no prospect of listing all such groups. However, if mp(G) ≥ 2, then the almost simple groups with a strongly p-embedded subgroup are known as a corollary to the classification of the finite simple groups. They include the rank 1 Lie type groups defined in characteristic p, and there are only additional examples if p ≤ 11. See Proposition 2.5 for a complete list. When we examine the list of simple groups with a strongly p-embedded subgroup H , we see that either H is a p-local subgroup or that F (H) ∼= Alt(p) × Alt(p) with p ≥ 5 or Ω8 (2) with p = 5. Thus the fact that H is strongly p-embedded severely restricts its structure. One major application of