Abstract
We determine the group of endotrivial modules (as an abstract group) for G a (quasi)simple group of sporadic type, extending previous results in the literature. In many sporadic cases we directly construct the subgroup of trivial-source endotrivial modules. We also resolve the question of whether certain simple modules for sporadic groups are endotrivial, posed by Lassueur, Malle and Schulte, in the majority of open cases. The results rely heavily on a recent description of the group of trivial-source endotrivial modules due to Grodal.
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Acknowledgements
I would like to thank: Jesper Grodal for some discussions about his at-the-time unpublished work, and for alerting me to some examples that he and his collaborators have found; Thomas Breuer and Richard Parker for help with computations with large modules for sporadic groups, whose contributions are detailed in the proof of Proposition 5.1; Gunter Malle for reading an early version of this manuscript; the referee for several helpful comments that have improved the paper’s exposition.
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Craven, D.A. Trivial-source endotrivial modules for sporadic groups. Beitr Algebra Geom 62, 317–343 (2021). https://doi.org/10.1007/s13366-020-00521-8
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DOI: https://doi.org/10.1007/s13366-020-00521-8