The classification of torsion endo-trivial modules

By Jon F. Carlson, Jacques Thévenaz. 2005

72 citations recorded by the source corpus.

The main purpose of this paper is to show that, if G is a finite p-group which is not cyclic, quaternion or semi-dihedral, then the group of endo-trivial G-modules is detected on restriction to elementary abelian p-subgroups of rank 2. Consequently, the torsion subgroup of the group of endo-trivial modules is trivial for any finite p-group, except in the known cases of cyclic, quaternion and semi-dihedral groups. This requires a large amount of group cohomology, recent bounds for the number of Bocksteins in Serre's theorem, Carlson's recent theorem expressing Serre's theorem in terms of modules, and finally the theory of varieties attached to modules. A detection theorem for the torsion subgroup of the Dade group of all endo-permutation modules is deduced from the main theorem. The structure of this torsion subgroup can then be fully described when p is odd (using the results of Bouc-Thevenaz).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d33c8c589e73dff99cafa4aaddf9d72a0acf2781

How researchers see this paper

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In Algebra and Number Theory

1. A. Mathew
2. Vesna Stojanoska
3. Paul Balmer
4. M. Schlichting
5. A. Elagin
6. Henning Krause
7. C. Walter
8. P. Seidel

In Discrete Mathematics and Combinatorics

1. A. Mathew
2. Paul Balmer

In Geometry and Topology

1. A. Mathew
2. Vesna Stojanoska
3. Paul Balmer
4. Lennart Meier
5. M. Schlichting
6. M. Hill
7. B. Bhatt
8. Peter Scholze

Citations in the corpus

• The Dade group of a p-group (2006, unknown): “Recently, in a series of remarkable papers ([13], [14], [17]), J. F. Carlson and J. Thévenaz completed the classification of endo-trivial modules, and proved detection theorems for the Dade group.”
• Stacks of group representations (2013, background): “The non-specialist will find in [Bal13, Bou06, CT04, CT05] further references on the central role played by endotrivial modules in modular representation theory.”
• The functor of units of Burnside rings for p-groups (2006, unknown): “Thévenaz ([13], Theorem 13.”
• The Picard group of topological modular forms via descent theory (2014, background): “The classification for a general p–group appears in [8].”