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

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.

Keywords

• brauer indecomposability
• Scott module
• generalized quaternion group

References

1. [1] M. Aschbacher, R. Kessar and B. Oliver, Fusion systems in algebra and topology, London Math. Soc. Lecture Series Notes 391, 2011.
2. [2] R. Brauer, Some applications of the theory of blocks of characters of finite groups II, J. Algebra 1, 307-334, 1964.
3. [3] C. Broto, R. Levi and B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16, 779-856, 2003.
4. [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. [5] M. Broué and L. Puig, Characters and local structure in G-algebras, J. Algebra 63, 306–317, 1980.
6. [6] D. Craven, The theory of fusion systems: an algebraic approach, Cambridge studies in advanced math. 131, Cambridge University Press, 2011.
7. [7] D. Craven and A. Glesser, Fusion systems on small p-groups, Trans. Amer. Math. Soc. 364 (11), 5945-5967, 2012.
8. [8] D. Gorenstein, Finite Groups, Harper and Row, New York, 1968.

9. [9] H. Ishioka and N. Kunugi. Brauer indecomposability of Scott modules, J. Algebra 470, 441-449, 2017.
10. [10] H. Kawai, On indecomposable modules and blocks, Osaka J. Math. 23, 201-205, 1986.
11. [11] R. Kessar, S. Koshitani and M. Linckelmann, On the Brauer indecomposability of Scott modules Quarterly J. Math. 66, 895-903, 2015.
12. [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. [13] S. Koshitani and C. Lassueur, Splendid Morita equivalences for principal 2-blocks with dihedral defect groups, Math. Z. 294, 639-666, 2020.
14. [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. [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. [16] S. Koshitani, İ. Tuvay, The Brauer indecomposability of Scott modules with semidihedral vertex, Proc. Edinb. Math. Soc. 64, 174–182, 2021.
17. [17] S. Koshitani, İ. Tuvay, The Brauer indecomposability of Scott modules with wreathed 2-group vertices, To appear in Rocky Mountain J. Math.
18. [18] M. Linckelmann, Introduction to fusion systems, Group Representation Theory, 79- 113, EPFL Press, Lausanne, 2007.
19. [19] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1989.
20. [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. [21] J. Thévenaz, G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995.
22. [22] İ. Tuvay, On Brauer indecomposability of Scott modules of Park-type groups, J. Group Theory 17, 1071-1079, 2014.