Endotrivial modules : a reduction to p′-central
extensions

Lassueur, Caroline; Thévenaz, Jacques

doi:10.2140/pjm.2017.287.423

Cited by 5 publications

(2 citation statements)

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“…• Finite groups of Lie type in type A in non-defining characteristic: Carlson-Mazza-Nakano in [24,25]. • A reduction to p -central extensions: Lassueur-Thévenaz in [56].…”

Section: Endo-trivial Modules Over Arbitrary Finite Groupsmentioning

confidence: 99%

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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“…• Finite groups of Lie type in type A in non-defining characteristic: Carlson-Mazza-Nakano in [24,25]. • A reduction to p -central extensions: Lassueur-Thévenaz in [56].…”

Section: Endo-trivial Modules Over Arbitrary Finite Groupsmentioning

confidence: 99%

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

Lassueur¹

2023

Jahresber. Dtsch. Math. Ver.

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“…1. The first one is a method we developed in [KL16] in order to treat finite groups with dihedral Sylow 2-subgroups, extended in [LT17a] to a more general method to relate the structure of T (G) to that of T (G/O p ′ (G)), which allows us to reduce the problem to groups with O 2 ′ (G) = 1. 2.…”

Section: Introductionmentioning

confidence: 99%

Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

Koshitani

Lassueur

2016

Journal of Group Theory

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Abstract. Let k be an algebraically closed field of characteristic p > 0 and G a finite group. We provide a description of the torsion subgroup T T (G) of the finitely generated abelian group T (G) of endo-trivial kG-modules when p = 2 and G has a dihedral Sylow 2-subgroup P . We prove that, in the case |P | ≥ 8, T T (G) ∼ = X(G) the group of one-dimensional kG-modules, except possibly when G/O 2 ′ (G) ∼ = A 6 , the alternating group of degree 6; in which case G may have 9-dimensional simple torsion endo-trivial modules. We also prove a similar result in the case |P | = 4, although the situation is more involved. Our results complement the tame-representation type investigation of endotrivial modules started by Carlson-Mazza-Thévenaz in the cases of semi-dihedral and generalized quaternion Sylow 2-subgroups. Furthermore we provide a general reduction result, valid at any prime p, to recover the structure of T T (G) from the structure of T T

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