The Chebotarev invariant of a finite group: A conjecture of Kowalski and Zywina

Lucchini, Andrea

doi:10.1090/proc/13805

The Chebotarev invariant of a finite group: A conjecture of Kowalski and Zywina

Author: Andrea Lucchini
Journal: Proc. Amer. Math. Soc. 146 (2018), 4549-4562
MSC (2010): Primary 20P05
DOI: https://doi.org/10.1090/proc/13805
Published electronically: August 10, 2018
MathSciNet review: 3856127
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A subset $\{g_1, \ldots , g_d\}$ of a finite group invariably generates if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates for every choice of $x_i \in G$ . The Chebotarev invariant of is the expected value of the random variable that is minimal subject to the requirement that randomly chosen elements of invariably generate . Confirming a conjecture of Kowalski and Zywina, we prove that there exists an absolute constant $\beta$ such that $C(G) \leq \beta \sqrt {\vert G\vert}$ for all finite groups

[1] M. Aschbacher and R. Guralnick, Some applications of the first cohomology group, J. Algebra 90 (1984), no. 2, 446–460. MR 760022, https://doi.org/10.1016/0021-8693(84)90183-2
[2] Adolfo Ballester-Bolinches and Luis M. Ezquerro, Classes of finite groups, Mathematics and Its Applications (Springer), vol. 584, Springer, Dordrecht, 2006. MR 2241927
[3] Eleonora Crestani and Andrea Lucchini, 𝑑-Wise generation of prosolvable groups, J. Algebra 369 (2012), 59–69. MR 2959786, https://doi.org/10.1016/j.jalgebra.2012.07.014
[4] E. Detomi and A. Lucchini, Crowns and factorization of the probabilistic zeta function of a finite group, J. Algebra 265 (2003), no. 2, 651–668. MR 1987022, https://doi.org/10.1016/S0021-8693(03)00275-8
[5] Eloisa Detomi and Andrea Lucchini, Invariable generation with elements of coprime prime-power orders, J. Algebra 423 (2015), 683–701. MR 3283736, https://doi.org/10.1016/j.jalgebra.2014.10.037
[6] John D. Dixon, Random sets which invariably generate the symmetric group, Discrete Math. 105 (1992), no. 1-3, 25–39. MR 1180190, https://doi.org/10.1016/0012-365X(92)90129-4
[7] Wolfgang Gaschütz, Praefrattinigruppen, Arch. Math. (Basel) 13 (1962), 418–426 (German). MR 146262, https://doi.org/10.1007/BF01650090
[8] Jason Fulman and Robert Guralnick, Derangements in simple and primitive groups, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 99–121. MR 1994962, https://doi.org/10.1142/9789812564481_0006
[9] Jason Fulman and Robert Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012), no. 6, 3023–3070. MR 2888238, https://doi.org/10.1090/S0002-9947-2012-05427-4
[10] Jason Fulman and Robert Guralnick, Derangements in subspace actions of finite classical groups, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2521–2572. MR 3592520, https://doi.org/10.1090/S0002-9947-2016-06721-5
[11] Jason Fulman and Robert Guralnick, Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture, Trans. Amer. Math. Soc. 370 (2018), no. 7, 4601–4622. MR 3812089, https://doi.org/10.1090/tran/7377
[12] Robert M. Guralnick, Generation of simple groups, J. Algebra 103 (1986), no. 1, 381–401. MR 860714, https://doi.org/10.1016/0021-8693(86)90194-8
[13] Robert M. Guralnick and Wolfgang Kimmerle, On the cohomology of alternating and symmetric groups and decomposition of relation modules, J. Pure Appl. Algebra 69 (1990), no. 2, 135–140. MR 1086556, https://doi.org/10.1016/0022-4049(90)90038-J
[14] Robert M. Guralnick and Corneliu Hoffman, The first cohomology group and generation of simple groups, Groups and geometries (Siena, 1996) Trends Math., Birkhäuser, Basel, 1998, pp. 81–89. MR 1644977, https://doi.org/10.1007/bf01214004
[15] Robert Guralnick, William M. Kantor, Martin Kassabov, and Alexander Lubotzky, Presentations of finite simple groups: profinite and cohomological approaches, Groups Geom. Dyn. 1 (2007), no. 4, 469–523. MR 2357481, https://doi.org/10.4171/GGD/22
[16] Paz Jiménez-Seral and Julio P. Lafuente, On complemented nonabelian chief factors of a finite group, Israel J. Math. 106 (1998), 177–188. MR 1656885, https://doi.org/10.1007/BF02773467
[17] W. M. Kantor, A. Lubotzky, and A. Shalev, Invariable generation and the Chebotarev invariant of a finite group, J. Algebra 348 (2011), 302–314. MR 2852243, https://doi.org/10.1016/j.jalgebra.2011.09.022
[18] Vicente Landazuri and Gary M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418–443. MR 360852, https://doi.org/10.1016/0021-8693(74)90150-1
[19] Martin W. Liebeck, Laszlo Pyber, and Aner Shalev, On a conjecture of G. E. Wall, J. Algebra 317 (2007), no. 1, 184–197. MR 2360145, https://doi.org/10.1016/j.jalgebra.2006.10.047
[20] Tomasz Łuczak and László Pyber, On random generation of the symmetric group, Combin. Probab. Comput. 2 (1993), no. 4, 505–512. MR 1264722, https://doi.org/10.1017/S0963548300000869
[21] Wolfgang Kimmerle, Richard Lyons, Robert Sandling, and David N. Teague, Composition factors from the group ring and Artin’s theorem on orders of simple groups, Proc. London Math. Soc. (3) 60 (1990), no. 1, 89–122. MR 1023806, https://doi.org/10.1112/plms/s3-60.1.89
[22] Emmanuel Kowalski and David Zywina, The Chebotarev invariant of a finite group, Exp. Math. 21 (2012), no. 1, 38–56. MR 2904906, https://doi.org/10.1080/10586458.2011.565261
[23] Andrea Lucchini and Gareth Tracey, An upper bound on the Chebotarev invariant of a finite group, Israel J. Math. 219 (2017), no. 1, 449–467. MR 3642029, https://doi.org/10.1007/s11856-017-1507-x
[24] Jitsuro Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28 (1952), 177–181. MR 50615
[25] Sheldon Ross, A first course in probability, 2nd ed., Macmillan Co., New York; Collier Macmillan Ltd., London, 1984. MR 732623
[26] Gary M. Seitz and Alexander E. Zalesskii, On the minimal degrees of projective representations of the finite Chevalley groups. II, J. Algebra 158 (1993), no. 1, 233–243. MR 1223676, https://doi.org/10.1006/jabr.1993.1132
[27] Pham Huu Tiep, Low dimensional representations of finite quasisimple groups, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 277–294. MR 1994973, https://doi.org/10.1142/9789812564481_0017