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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Endotrivial modules for finite groups via homotopy theory
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by Jesper Grodal
J. Amer. Math. Soc. 36 (2023), 177-250
Published electronically: April 21, 2022


Classifying endotrivial $kG$-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group $G$, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thévenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module $k$ direct sum a projective module when restricted to a Sylow $p$-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivial $p$-subgroups with values in the units $k^\times$, viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers in $G$, in particular verifying the Carlson–Thévenaz conjecture—this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of the $p$-subgroup complex and $p$-fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.
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Bibliographic Information
  • Jesper Grodal
  • Affiliation: Department of Mathematical Sciences, University of Copenhagen, Denmark
  • MR Author ID: 634219
  • ORCID: 0000-0002-2901-8525
  • Email:
  • Received by editor(s): February 1, 2019
  • Received by editor(s) in revised form: February 24, 2020, June 24, 2020, June 25, 2020, July 16, 2021, July 20, 2021, and July 22, 2021
  • Published electronically: April 21, 2022
  • Additional Notes: This work was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92 and DNRF151). The author enjoyed the hospitality of MSRI Berkeley, Spring 2018 (NSF grant DMS-1440140) and the Isaac Newton Institute, Cambridge, Fall 2018 (EPSRC grants EP/K032208/1 and EP/R014604/1) where the manuscript was revised
  • © Copyright 2022 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 36 (2023), 177-250
  • MSC (2020): Primary 20C20; Secondary 20J05, 55P91
  • DOI:
  • MathSciNet review: 4495841