Strongly p-embedded Subgroups

For p a prime and G a finite group, a proper subgroup H of G is strongly p-embedded in G if and only if p divides vertical bar H vertical bar but does not divide broken vertical bar H boolean AND H-g broken vertical bar for all g is an element of G \ H. This article contributes to the various projects to improve the classification of the finite simple groups. One of the main theorems is as follows: Suppose that G is a finite group in which every proper subgroup has composition factors from the known simple groups, p is an odd prime and that H is a strongly p-embedded subgroup of G. Assume that H boolean AND K is of even order for all non-trivial normal subgroups K of G, O-p' (H) = 1 and m(p) (C-H(t)) >= 2 for every involution t of H. Then there exists n >= 2 such that either F* (G) congruent to PSU3 (p(n)) or p = 3 and F* (G) congruent to 2G(2) (3(2n-1)).