Step - by - Step Conjugation of p - Subgroups of a Group

Alperin [1] has recently introduced a fundamental method for conjugating from one p-Sylow subgroup Q of a finite group G to a second p-Sylow subgroup P in a series of steps. The importance of this is that it provides information concerning the conjugacy in G of subsets of P, that is, fusion of subsets of P. In certain situations it is possible to extend these results to the case of conjugate p-subgroups Q and P of G which are not p-Sylow subgroups of G. The situation of prime importance is that of a p-Sylow subgroup P of a subgroup of G. That is, P is a p-Sylow subgroup of the stabilizer of a point in transitive permutation representation of G. For example, our results apply to the case of a p-Sylow subgroup of the stabilizer of a point in a 2-transitive permutation group. In general, we are able to handle groups having a sufficiently tight structure, such as groups having a split (B, N)-pair [4]. Moreover, solvable groups can be constructed in which the type of step-by-step conjugation we consider need not apply to p-subgroups other than p-Sylow subgroups. When P is a p-Sylow subgroup of G, Alperin [I] has indicated many applications of his results to transfer. As we are concerned with p-subgroups that are not necessarily p-Sylow subgroups, our results seem most applicable to fusion. In fact, the significance of our results on conjugation is that they lead in a natural way to new results on fusion. \Ve were thus led in §3 to possibly unexpected results on fusion; most of these can, however, be easily proved by direct methods. Our notation is all standard. A C B denotes that A is a subset of B, while but A=/:. B. If A is a subset of a group G, NG(A) is the normalizer of A in G and CG(A) is the centralizer of A in G. All groups considered will be finite. We are grateful to I. M. Isaaes for making several helpful suggestions and for providing Lemma 3.8.