On Brauer stars

Abstract

This paper contains some results on how representations in the principal p-block of a finite group with a cyclic Sylow p-subgroup yield information on the structure of the group. Throughout the paper, G denotes a finite group, p a fixed odd prime dividing 1 GI, P a Sylow p-subgroup of G with ]P] = p”, and B,(p) the principal p-block of G. If P is cyclic, then the ordinary and modular irreducible characters in any p-block of G can be described in terms of a graph, called the Bruuer tree belonging to the block (as in [5, VII.61). A tree is called a star if there are no paths with more than two edges, i.e.,