The classification of torsion endo-trivial modules

Carlson, Jon; Thévenaz, Jacques

doi:10.4007/annals.2005.162.823

Cited by 72 publications

(154 citation statements)

References 15 publications

Supporting

Mentioning

154

Contrasting

Order By: Relevance

“…In this paper we show this is true for p d 5. Our first result is a consequence of a result of Carlson and Thévenaz [2]. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

Endo-permutation modules arising from the action of a p-group on a defect zero block

Salminen¹

2009

Journal of Group Theory

View full text Add to dashboard Cite

Abstract. Let p be an odd prime and let k be an algebraically closed field of characteristic p. Also, let G be a finite p 0 -group. By Maschke's theorem, kG is isomorphic to a product Q t i¼1 End k ðV i Þ as a k-algebra. Suppose that a p-subgroup P of AutðGÞ stabilizes End k ðV i0 Þ for some i 0 . Such a V i0 will be an endo-permutation kP-module. Puig showed that the only modules that occur in this way are those whose image is torsion in the Dade group DðPÞ.If G is any finite group and b is a defect zero block of kG, then kGb G End k ðLÞ for some L. If kGb is P-stable for some p-subgroup P of AutðGÞ and Br P ðbÞ 0 0, then L will again be an endo-permutation kP-module. We show that if p d 5, then L is torsion in DðPÞ. This result depends on the classification of the finite simple groups.

show abstract

“…In this paper we show this is true for p d 5. Our first result is a consequence of a result of Carlson and Thévenaz [2]. Theorem 1.2.…”

Section: Introductionmentioning

confidence: 89%

“…Since Q=R is a cyclic group, V is isomorphic to V Ã as a kQ=R-modules. But this implies that V and V Ã are isomorphic as kQ-modules, and self-dual Q-modules are torsion in DðQÞ from [2]. Therefore the action of Q on H must be faithful.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Endo-permutation modules arising from the action of a p-group on a defect zero block

Salminen¹

2009

Journal of Group Theory

View full text Add to dashboard Cite

show abstract

“…It is proved in [12] (using also [13]), but the statement now incorporates the improvement obtained in Theorem 3.1.…”

Section: Construction Via Relative Syzygiesmentioning

confidence: 99%

Torsion-free endotrivial modules

2014

Self Cite

View full text Add to dashboard Cite

Abstract. Let G be a finite group and let T (G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We investigate the torsion-free part T F (G) of the group T (G) and look for generators of T F (G). We describe three methods for obtaining generators. Each of them only gives partial answers to the question but we obtain more precise results in some specific cases. We also conjecture that T F (G) can be generated by modules belonging to the principal block and we prove the conjecture in some cases.

show abstract

“…For D tors , a detection theorem from [CaTh1] together with a result from [BoTh2] about elementary abelian groups imply that the unit morphism…”

Section: Theorem Let P Be An Odd Prime Number and Let D Be The Dade mentioning

confidence: 99%

“…The set of equivalence classes of such modules is an abelian group D(P ) (with respect to tensor product), called the Dade group of P . An important ingredient for the classification is a detection theorem, proved in [CaTh1], which asserts (for odd p) that the product of all deflation-restriction maps …”

Section: Introductionmentioning

confidence: 99%

A sectional characterization of the Dade group

Bouc¹,

Thévenaz²

2008

Journal of Group Theory

Self Cite

View full text Add to dashboard Cite

: Let k be a field of characteristic p , let P be a finite p-group, where p is an odd prime, and let D(P ) be the Dade group of endo-permutation kP -modules. It is known that D(P ) is detected via deflation-restriction by the family of all sections of P which are elementary abelian of rank ≤ 2. In this paper, we improve this result by characterizing D(P ) as the limit (with respect to deflationrestriction maps and conjugation maps) of all groups D(T /S) where T /S runs through all sections of P which are either elementary abelian of rank ≤ 3 or extraspecial of order p 3 and exponent p.

show abstract

The classification of torsion endo-trivial modules

Cited by 72 publications

References 15 publications

Endo-permutation modules arising from the action of a p-group on a defect zero block

Endo-permutation modules arising from the action of a p-group on a defect zero block

Torsion-free endotrivial modules

A sectional characterization of the Dade group

Contact Info

Product

Resources

About