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depth of group algebras; finite group; faithful representation
This paper gives necessary and sufficient conditions for subgroups with trivial core to be of odd depth. We show that a subgroup with trivial core is an odd depth subgroup if and only if certain induced modules from it are faithful. Algebraically this gives a combinatorial condition that has to be satisfied by the subgroups with trivial core in order to be subgroups of a given odd depth. The condition can be expressed as a certain matrix with $\{0,1\}$-entries to have maximal rank. The entries of the matrix correspond to the sizes of the intersections of the subgroup with any of its conjugate.
[1] Boltje, R., Danz, S., Külshammer, B.: On the depth of subgroups and group algebra extensions. J. Algebra 335 (2011), 258-281. DOI 10.1016/j.jalgebra.2011.03.019 | MR 2792576 | Zbl 1250.20001
[2] Boltje, R., Külshammer, B.: On the depth $2$ condition for group algebra and Hopf algebra extensions. J. Algebra 323 (2010), 1783-1796. DOI 10.1016/j.jalgebra.2009.11.043 | MR 2588139 | Zbl 1200.16035
[3] Bourgain, J., Vu, V. H., Wood, P. M.: On the singularity probability of discrete random matrices. J. Funct. Anal. 258 (2010), 559-603. DOI 10.1016/j.jfa.2009.04.016 | MR 2557947 | Zbl 1186.60003
[4] Burciu, S., Kadison, L.: Subgroups of depth three. Perspectives in Mathematics and Physics: Essays Dedicated to Isadore Singer's 85th Birthday T. Mrowka et al. Surveys in Differential Geometry 15 International Press, Somerville (2011), 17-36. MR 2815724 | Zbl 1242.16028
[5] Burciu, S., Kadison, L., Külshammer, B.: On subgroup depth. (With an appendix by S. Danz and B. Külshammer). Int. Electron. J. Algebra (electronic only) 9 (2011), 133-166. MR 2753764 | Zbl 1266.20001
[6] Gantmakher, F. R.: Matrix theory. With an appendix by V. B. Lidskij. With a preface by D. P. Želobenko. Translated from the second Russian edition by H. Boseck, D. Soyka and K. Stengert, Hochschulbücher für Mathematik, Bd. 86 VEB Deutscher Verlag der Wissenschaften, Berlin (1986), German. MR 0869996
[7] Goodman, F. M., Harpe, P. De la, Jones, V. F. R.: Coxeter Graphs and Towers of Algebras. Mathematical Sciences Research Institute Publications 14 Springer, New York (1989). DOI 10.1007/978-1-4613-9641-3 | MR 0999799 | Zbl 0698.46050
[8] Kadison, L., Külshammer, B.: Depth two, normality and a trace ideal condition for Frobenius extensions. Commun. Algebra 34 (2006), 3103-3122. DOI 10.1080/00927870600650291 | MR 2252660 | Zbl 1115.16020
[9] Metropolis, N., Stein, P. R.: On a class of $(0,1)$ matrices with vanishing determinants. J. Comb. Theory. 3 (1967), 191-198. DOI 10.1016/S0021-9800(67)80006-1 | MR 0211889 | Zbl 0153.02301
[10] Rieffel, M. A.: Normal subrings and induced representations. J. Algebra 59 (1979), 364-386. DOI 10.1016/0021-8693(79)90133-9 | MR 0543256 | Zbl 0496.16035
[11] Živković, M.: Classification of small $(0,1)$ matrices. Linear Algebra Appl. 414 (2006), 310-346. DOI 10.1016/j.laa.2005.10.010 | MR 2209249 | Zbl 1091.15022
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