Abstract
Let
Funding statement: The author is a member of INdAM research group for Algebraic and Geometric Structures and their Applications (GNSAGA). He is also a member of the research training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology, funded by DFG. The author thanks both groups for supporting this project.
Acknowledgements
I want to thank Francesca Dalla Volta for her careful supervision during the preparation of my Ph.D. thesis, from which this work has been extracted.
Communicated by: Andrea Lucchini
References
[1] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469โ514. 10.1007/BF01388470Search in Google Scholar
[2] L. Brickman and P.โA. Fillmore, The invariant subspace lattice of a linear transformation, Canadian J. Math. 19 (1967), 810โ822. 10.4153/CJM-1967-075-4Search in Google Scholar
[3] V. Colombo and A. Lucchini, On subgroups with non-zero Mรถbius numbers in the alternating and symmetric groups, J. Algebra 324 (2010), no. 9, 2464โ2474. 10.1016/j.jalgebra.2010.07.040Search in Google Scholar
[4] F. Dalla Volta and L. Di Gravina, Mรถbius function of the subgroup lattice of a finite group and Euler characteristic, preprint (2022), https://arxiv.org/abs/2212.01917. Search in Google Scholar
[5] C.โD. Godsil, An introduction to the Moebius function, preprint (2018), https://arxiv.org/abs/1803.06664. Search in Google Scholar
[6] P. Hall, The Eulerian functions of a group, Quart. J. Math. 7 (1936), 134โ151. 10.1093/qmath/os-7.1.134Search in Google Scholar
[7] T. Hawkes, I.โM. Isaacs and M. รzaydin, On the Mรถbius function of a finite group, Rocky Mountain J. Math. 19 (1989), no. 4, 1003โ1034. 10.1216/RMJ-1989-19-4-1003Search in Google Scholar
[8] L. Kalmar, รber die mittlere Anzahl der Produktdarstellungen der Zahlen. (Erste Mitteilung), Acta Litterarum Sci. (Szeged) 5 (1931), 95โ107. Search in Google Scholar
[9] P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Math. Soc. Lecture Note Ser. 129, Cambridge University, Cambridge, 1990. 10.1017/CBO9780511629235Search in Google Scholar
[10] C. Kratzer and J. Thรฉvenaz, Fonction de Mรถbius dโun groupe fini et anneau de Burnside, Comment. Math. Helv. 59 (1984), no. 3, 425โ438. 10.1007/BF02566359Search in Google Scholar
[11] A. Lucchini, On the subgroups with non-trivial Mรถbius number, J. Group Theory 13 (2010), no. 4, 589โ600. 10.1515/jgt.2010.009Search in Google Scholar
[12] A. Mann, A probabilistic zeta function for arithmetic groups, Internat. J. Algebra Comput. 15 (2005), no. 5โ6, 1053โ1059. 10.1142/S0218196705002633Search in Google Scholar
[13] P.โM. Neumann and C.โE. Praeger, Cyclic matrices in classical groups over finite fields, J. Algebra 234 (2000), 367โ418. 10.1006/jabr.2000.8548Search in Google Scholar
[14] R.โP. Stanley, Enumerative Combinatorics. Vol. I, 2nd ed., Cambridge University, Cambridge, 2012. 10.1017/CBO9781139058520Search in Google Scholar
[15] R.โA. Wilson, The Finite Simple Groups, Grad. Texts in Math. 251, Springer, London, 2009. 10.1007/978-1-84800-988-2Search in Google Scholar
ยฉ 2023 Walter de Gruyter GmbH, Berlin/Boston