Transfer and fusion in finite groups

Abstract

There exist a number of well-known theorems which give conditions under which the structure of the normalizer of a particular p-subgroup of a finite group G determines certain “global” properties of G, such as, the largest abelian p-factor group of G or the conjugacy of p-elements in G. For example, we may mention Burnside’s theorems on the conjugacy of elements in the center of a Sylow p-subgroup and on the existence of normal p-complements in groups with abelian Sylow p-subgroups. The theorem of Griin concerning p-normality and the Hall-Wielandt theorem are also of this nature. In this paper we shall establish a number of general results of this type. To state these, we must first introduce several concepts. Throughout the paper G will denote a fixed finite group, p will be a fixed prime divisor of the order of G, and Z will designate the set of all nonidentity p-subgroups of G.