Sylow Intersections and Fusion

It is common in mathematics for a subject to have its local and global aspects; such is the case in group theory. For example, the structure and embedding of subgroups of a group G may be usefully thought of as part of the local structure of G while the normal subgroups, quotient groups and conjugacy classes are relevant to the global structure of G. Furthermore, the connections between local and global structure are very important. In the study of these relations, the methods of representation theory and transfer are very useful. The application of these techniques is often based upon results concerning the fusion of elements. (Recall that two elements of a subgroup H of a group G are said to be ficsed if they are conjugate in G but not in H.) Indeed, the formula for induced characters clearly illustrates this dependence. However, more pertinent to the present work, and also indicative of this connection with fusion, is the focal subgroup theorem [8]: if P is a Sylow p-subgroup of a group G then P n G’ is generated by all elements of the form a-lb, where a and b are elements of P conjzgate in G. Hence, this result, an application of transfer, shows that the fusion of elements of P determines P n G’ and thus P/P n G’ which is isomorphic with the largest Abelian p-quotient group of G. It is the purpose of this paper to demonstrate that the fusion of elements of a Sylow subgroup P is completely determined by the normalizers of the nonidentity subgroups of P. Therefore, P/P n G’, a global invariant of G, is completely described by the local structure of G. A weak form of our main result is as follows : if a and b are elements of a Sylow subgroup P of the group G and a and b are conjugate in G, then there exist elements a, ,..., a,,, of P and subgroups HI ,..., H,, of P such that a = a, , b = a, and ai and U~+~ are contained in Hi and cmjugate in N(H,), 1 < i < m 1. We shall strengthen