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  • Print publication year: 2011
  • Online publication date: September 2011

Introduction

Summary

Let G be a finite group, p a prime, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate under some element of G. The term “fusion” seems to have been introduced by Brauer in the fifties, but the general notion has been of interest for over a century. For example, in his text The Theory of Groups of Finite Order [Bu] (first published in 1897), Burnside proved that if S is abelian then the normalizer in G of S controls fusion in S. (A subgroup H of G is said to control fusion in S if any pair of tuples of elements of S which are conjugate in G are also conjugate under H.)

Initially, information about fusion was usually used in conjunction with transfer, as in the proof of the normal p-complement theorems of Burnside and Frobenius, which showed that, under suitable hypotheses on fusion, G possesses a normal p-complement: a normal subgroup of index |S| in G. But in the sixties and seventies more sophisticated results on fusion began to appear, such as Alperin's Fusion Theorem [Al1], which showed that the family of normalizers of suitable subgroups of S control fusion in S, and Goldschmidt's Fusion Theorem [Gd3], which determined the groups G possessing a nontrivial abelian subgroup A of S such that no element of A is fused into S\A.

In the early nineties, Lluis Puig abstracted the properties of G-fusion in a Sylow subgroup S, in his notion of a Frobenius category on a finite p-group S, by discarding the group G and focusing instead on isomorphisms between subgroups of S.