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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 5

The Brauer indecomposability of Scott modules with vertex $Q_{2^n}\times C_{2^m}$

İpek Tuvay(MIMAR SINAN FINE ARTS UNIVERSITY, FACULTY OF SCIENCE AND LETTERS)0000-0002-7427-9310
Pages
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Publication Date
2021-10-15
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

This result generalizes similar results which are obtained for abelian, dihedral, generalized quaternion, semidihedral and wreathed $2$-group vertices.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We prove that the Scott module whose vertex is isomorphic to a direct product of a generalized quaternion $2$-group and a cyclic $2$-group is Brauer indecomposable.
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Key Findings & Data:We prove that the Scott module whose vertex is isomorphic to a direct product of a generalized quaternion $2$-group and a cyclic $2$-group is Brauer indecomposable.
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Conclusion & Impact:This result generalizes similar results which are obtained for abelian, dihedral, generalized quaternion, semidihedral and wreathed $2$-group vertices.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:brauer indecomposability · Scott module · generalized quaternion group

Abstract

We prove that the Scott module whose vertex is isomorphic to a direct product of a generalized quaternion $2$-group and a cyclic $2$-group is Brauer indecomposable. This result generalizes similar results which are obtained for abelian, dihedral, generalized quaternion, semidihedral and wreathed $2$-group vertices.

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Cite This Article

Tuvay, İ. (2021). The Brauer indecomposability of Scott modules with vertex $Q_{2^n}\times C_{2^m}$. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 5. https://doi.org/10.15672/hujms.815694

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References (22)

  1. M. Aschbacher, R. Kessar and B. Oliver, Fusion systems in algebra and topology, London Math. Soc. Lecture Series Notes 391, 2011.
  2. R. Brauer, Some applications of the theory of blocks of characters of finite groups II, J. Algebra 1, 307-334, 1964.
  3. C. Broto, R. Levi and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16, 779-856, 2003.
  4. M. Broué, On Scott modules and p-permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93, 401-408, 1985.
  5. M. Broué and L. Puig, Characters and local structure in G-algebras, J. Algebra 63, 306–317, 1980.
  6. D. Craven, The theory of fusion systems: an algebraic approach, Cambridge studies in advanced math. 131, Cambridge University Press, 2011.
  7. D. Craven and A. Glesser, Fusion systems on small p-groups, Trans. Amer. Math. Soc. 364 (11), 5945-5967, 2012.
  8. D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.
  9. H. Ishioka and N. Kunugi. Brauer indecomposability of Scott modules, J. Algebra 470, 441-449, 2017.
  10. H. Kawai, On indecomposable modules and blocks, Osaka J. Math. 23, 201-205, 1986.
  11. R. Kessar, S. Koshitani and M. Linckelmann, On the Brauer indecomposability of Scott modules Quarterly J. Math. 66, 895-903, 2015.
  12. R. Kessar, N. Kunugi and N. Mitsuhashi, On saturated fusion systems and Brauer indecomposability of Scott modules, J. Algebra 340, 90-103, 2011.
  13. S. Koshitani and C. Lassueur, Splendid Morita equivalences for principal 2-blocks with dihedral defect groups, Math. Z. 294, 639-666, 2020.
  14. S. Koshitani and C. Lassueur, Splendid Morita equivalences for principal 2-blocks with generalised quaternion defect groups J. Algebra 558, 523-533, 2020.
  15. S. Koshitani and İ. Tuvay, The Brauer indecomposability of Scott modules for the quadratic group Qd(p), Algebr. Represent. Theor. 22, 1387-1397, 2019.
  16. S. Koshitani, İ. Tuvay, The Brauer indecomposability of Scott modules with semidihedral vertex, Proc. Edinb. Math. Soc. 64, 174–182, 2021.
  17. S. Koshitani, İ. Tuvay, The Brauer indecomposability of Scott modules with wreathed 2-group vertices, To appear in Rocky Mountain J. Math.
  18. M. Linckelmann, Introduction to fusion systems, Group Representation Theory, 79- 113, EPFL Press, Lausanne, 2007.
  19. H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1989.
  20. B. Sambale, Blocks with defect group $Q_{2^n}\times C_{2^m}$ and $SD_{2^n}\times C_{2^m}$, Algebr. Repr.Theor. 16, 1717-1732, 2013.
  21. J. Thévenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995.
  22. İ. Tuvay, On Brauer indecomposability of Scott modules of Park-type groups, J. Group Theory 17, 1071-1079, 2014.