Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

Abstract

We finish off the classification of the endo-trivial modules of finite groups with Sylow 2-subgroups isomorphic to a semi-dihedral 2-group started by Carlson, Mazza and Thévenaz in their article Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup published in 2013.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alperin, J.L., Brauer, R., Gorenstein, D.: Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups. Trans. Am. Math. Soc. 151, 1–261 (1970)

    MathSciNet  MATH  Google Scholar 

  • Bender, H.: Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt. J. Algebra 17, 527–554 (1971)

    Article  MathSciNet  Google Scholar 

  • Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)

    Article  MathSciNet  Google Scholar 

  • Carlson, J.F., Thévenaz, J.: Torsion endo-trivial modules. Algebra Represent. Theory 3, 303–335 (2000)

    Article  MathSciNet  Google Scholar 

  • Carlson, J.F., Mazza, N., Thévenaz, J.: Endotrivial modules for \(p\)-solvable groups. Trans. Am. Math. Soc. 363, 4979–4996 (2011)

    Article  MathSciNet  Google Scholar 

  • Carlson, J.F., Mazza, N., Thévenaz, J.: Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup. J. Eur. Math. Soc. (JEMS) 15, 157–177 (2013)

    Article  MathSciNet  Google Scholar 

  • Carlson, J.F., Mazza, N., Nakano, D.K.: Endotrivial modules for finite groups of Lie type \(A\) in nondefining characteristic. Math. Z. 282, 1–24 (2016)

    Article  MathSciNet  Google Scholar 

  • Carter, R., Fong, P.: The Sylow 2-subgroups of the finite classical groups. J. Algebra 1, 139–151 (1964)

    Article  MathSciNet  Google Scholar 

  • Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985)

    MATH  Google Scholar 

  • Craven, D.A.: Trivial-source endotrivial modules for sporadic groups. Beitr. Algebra Geom. (2020). https://doi.org/10.1007/s13366-020-00521-8

    Article  MATH  Google Scholar 

  • Dade, E.C.: Endo-permutation modules over p-groups. I. Ann. Math. 107, 459–494 (1978)

    Article  MathSciNet  Google Scholar 

  • Dade, E.C.: Endo-permutation modules over p-groups. II. Ann. Math. 108, 317–346 (1978)

    Article  MathSciNet  Google Scholar 

  • Gorenstein, D.: Finite groups the centralizers of whose involutions have normal 2-complements. Can. J. Math. 21, 335–357 (1969)

    Article  MathSciNet  Google Scholar 

  • Grodal, J.: Endotrivial modules for finite groups via homotopy theory. arXiv:1608.00499v2 (2018)

  • Isaacs, I.M.: Finite Group Theory, Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence (2008)

    Google Scholar 

  • Jones, M.R.: Some inequalities for the multiplicator of a finite group. II. Proc. Am. Math. Soc. 45, 167–172 (1974)

    Article  MathSciNet  Google Scholar 

  • Karpilovsky, G.: The Schur multiplier, London Mathematical Society Monographs New Series, vol. 2. The Clarendon Press, Oxford University Press, New York (1987)

    Google Scholar 

  • Kawata, S.: On Auslander–Reiten components for certain group modules. Osaka J. Math. 30, 137–157 (1993)

    MathSciNet  MATH  Google Scholar 

  • Kondrat’ev, A.S.: Normalizers of Sylow 2-subgroups in finite simple groups. Mat. Zametki 78, 368–376 (2005)

    Article  MathSciNet  Google Scholar 

  • Koshitani, S., Lassueur, C., Sambale, B.: Splendid Morita equivalences for principal blocks with semidihedral defect groups. arXiv:2010.08541 (2020)

  • Koshitani, S., Lassueur, C.: Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup. J. Group Theory 19, 635–660 (2016)

    MathSciNet  MATH  Google Scholar 

  • Lassueur, C., Mazza, N.: Endotrivial modules for the sporadic groups and their covers. J. Pure Appl. Algebra 219, 4203–4228 (2015)

    Article  MathSciNet  Google Scholar 

  • Lassueur, C., Thévenaz, J.: Endotrivial modules: a reduction to \(p^{\prime }\)-central extensions. Pac. J. Math. 287, 423–438 (2017)

    Article  MathSciNet  Google Scholar 

  • Lassueur, C., Thévenaz, J.: Universal \(p^{\prime }\)-central extensions. Expos. Math. 35, 237–251 (2017)

    Article  MathSciNet  Google Scholar 

  • Mazza, N.: Endotrivial Modules. Springer Briefs in Mathematics. Springer, Cham (2019)

    Book  Google Scholar 

  • Mazza, N., Thévenaz, J.: Endotrivial modules in the cyclic case. Archiv der Mathematik 89, 497–503 (2007)

    Article  MathSciNet  Google Scholar 

  • Schaeffer Fry, A.A., Taylor, J.: On self-normalising Sylow 2-subgroups in type A. J. Lie Theory 28, 139–168 (2018)

    MathSciNet  MATH  Google Scholar 

  • Tahara, K.I.: On the second cohomology groups of semidirect products. Math. Z. 129, 365–379 (1972)

    Article  MathSciNet  Google Scholar 

  • Wilson, R., Thackray, J., Parker, R., Noeske, F., Müller, J., Lübeck, F., Jansen, C., Hiss, G., Breuer, T.: The modular atlas project. http://www.math.rwth-aachen.de/ MOC

Download references

Acknowledgements

The authors would like to thank Jesper Grodal, Burkhard Külshammer, Gunter Malle, Nadia Mazza, Benjamin Sambale, Mandi Schaeffer Fry, Gernot Stroth, Jacques Thévenaz, and Rebecca Waldecker for helpful hints and discussions related to the content of this article. They are grateful to the referee for detailed comments and suggestions.

Author information

Authors and Affiliations

  1. Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

    Shigeo Koshitani

  2. FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Caroline Lassueur

Authors
  1. Shigeo Koshitani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Caroline Lassueur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Caroline Lassueur.

Additional information

Dedicated to Jon Carlson on the occasion of his 80th Birthday and to Jacques Thévenaz on the occasion of his 70th Birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C) 19K03416, 2019–2021. The second author acknowledges financial support by DFG SFB/TRR 195. This piece of work is part of Project A18 thereof.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Koshitani, S., Lassueur, C. Trivial source endo-trivial modules for finite groups with semi-dihedral Sylow 2-subgroups. Beitr Algebra Geom 63, 233–246 (2022). https://doi.org/10.1007/s13366-021-00581-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-021-00581-4

Keywords

Mathematics Subject Classification

Search

Navigation