Trivial source endo-trivial modules for finite groups with semi-dihedral Sylow 2-subgroups

Koshitani, Shigeo; Lassueur, Caroline

doi:10.1007/s13366-021-00581-4

Cited by 1 publication

(2 citation statements)

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“…• Groups with a semi-dihedral Sylow 2-subgroup: Carlson-Mazza-Thévenaz in [27] and Koshitani-Lassueur in [46]. • Finite groups of Lie type in non-defining characteristic, torsion-free part only:…”

Section: Endo-trivial Modules Over Arbitrary Finite Groupsmentioning

confidence: 99%

“…The aforementioned results led to a classification of the simple endo-trivial modules for the finite quasi-simple groups based on ordinary character theory in [52,53,62,63], partially involving computer algebra methods and computation with GAP4, CHEVIE, or MAGMA. Character-theoretical arguments were also used to determine the structure of the group T k (G) of endo-trivial modules, and in particular of its torsion subgroup T tors k (G), for certain finite quasi-simple groups of Lie type in [52], for the sporadic groups and their covering groups in [55], for the Schur covers of the alternating and symmetric groups in [54], or four some groups with dihedral or semi-dihedral Sylow 2-subgroups in [44][45][46].…”

Section: Lifting Endo-trivial Kg-modulesmentioning

confidence: 99%

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A Tour of $p$-Permutation Modules and Related Classes of Modules

Lassueur¹

2023

Jahresber. Dtsch. Math. Ver.

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This survey provides an overview of numerous results on $p$ p -permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-$p$ p -permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the $p$ p -permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.

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Section: Endo-trivial Modules Over Arbitrary Finite Groupsmentioning

confidence: 99%