Autoequivalences of blocks and a conjecture of Zassenhaus

Abstract

In this paper, we show that for every finite group with cyclic Sylow p-subgroups the principal p-block B is rigid with respect to the trivial simple module. This means that each autoequivalence which fixes the trivial simple module fixes the isomorphism class of each finitely generated B-module. As a consequence each augmentation preserving automorphism of the integral group ring of PSL(2, p), p a rational prime, is given by a group automorphism followed by a conjugation in QPSL(2, p). In particular this proves a conjecture of Zassenhaus for these groups. Finally we show the same statement for a couple of other simple groups by different methods.