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Autoequivalences of blocks and a conjecture of Zassenhaus
Journal article   Open access   Peer reviewed

Autoequivalences of blocks and a conjecture of Zassenhaus

Frauke M Bleher, Gerhard Hiss and Wolfgang Kimmerle
Journal of pure and applied algebra, Vol.103(1), pp.23-43
1995
DOI: 10.1016/0022-4049(94)00091-V
url
https://doi.org/10.1016/0022-4049(94)00091-VView
Published (Version of record) Open Access

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.

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