Walking around the Brauer tree

Abstract

Let G be a finite group, and k a field of finite characteristic p , such that the polynomial x ¦ G ¦ –1 splits completely in k [ x ]. Let Β be a kG -block which has defect group D which is cylclic of order p d ( d ≧ 1). Brauer showed in a famous paper [2] that, in case d = 1, the decomposition matrix of Β is determined by a certain positive integer e which divides p − 1, and a tree Г, a connected acyclic linear graph of e + 1 vertices and e edges. Twenty-five years later Dade ([3]) extended Brauer's theorem to the general case.