Conjugacy classes of derangements in finite groups of Lie type

Eberhard, Sean; Garzoni, Daniele

doi:10.1090/btran/193

Conjugacy classes of derangements in finite groups of Lie type
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by Sean Eberhard and Daniele Garzoni;

Trans. Amer. Math. Soc. Ser. B 12 (2025), 536-575

DOI: https://doi.org/10.1090/btran/193

Published electronically: April 22, 2025

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Abstract:

Let $G$ be a finite almost simple group of Lie type acting faithfully and primitively on a set $\Omega$. We prove an analogue of the Boston–Shalev conjecture for conjugacy classes: the proportion of conjugacy classes of $G$ consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).

References

M. Aschbacher, Finite group theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008, DOI 10.1017/CBO9781139175319
M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
Omran Ahmadi and Gerardo Vega, On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields, Finite Fields Appl. 14 (2008), no. 1, 124–131. MR 2381481, DOI 10.1016/j.ffa.2006.09.004
L. Carlitz, Some theorems on irreducible reciprocal polynomials over a finite field, J. Reine Angew. Math. 227 (1967), 212–220. MR 215815, DOI 10.1515/crll.1967.227.212
Peter J. Cameron and Arjeh M. Cohen, On the number of fixed point free elements in a permutation group, Discrete Math. 106/107 (1992), 135–138. A collection of contributions in honour of Jack van Lint. MR 1181907, DOI 10.1016/0012-365X(92)90540-V
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
Stephen D. Cohen, On irreducible polynomials of certain types in finite fields, Proc. Cambridge Philos. Soc. 66 (1969), 335–344. MR 244202, DOI 10.1017/s0305004100045023
Sean Eberhard, Kevin Ford, and Ben Green, Permutations fixing a $k$-set, Int. Math. Res. Not. IMRN 21 (2016), 6713–6731. MR 3579977, DOI 10.1093/imrn/rnv371
P. Erdős and M. Szalay, On some problems of J. Dénes and P. Turán, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 187–212. MR 820222
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, DOI 10.1142/9789812564481_{0}006
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, DOI 10.1090/S0002-9947-2012-05427-4
Jason Fulman and Robert Guralnick, The number of regular semisimple conjugacy classes in the finite classical groups, Linear Algebra Appl. 439 (2013), no. 2, 488–503. MR 3089699, DOI 10.1016/j.laa.2013.03.031
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, DOI 10.1090/tran/6721
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, DOI 10.1090/tran/7377
Kevin Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. (2) 168 (2008), no. 2, 367–433. MR 2434882, DOI 10.4007/annals.2008.168.367
Kevin Ford, Cycle type of random permutations: a toolkit, Discrete Anal. (2022), Paper No. 9, 36. MR 4481406
Jason Fulman, Cycle indices for the finite classical groups, J. Group Theory 2 (1999), no. 3, 251–289. MR 1696313, DOI 10.1515/jgth.1999.017
Patrick X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175–179. MR 276318, DOI 10.1007/BF01113339
Robert M. Guralnick and Frank Lübeck, On $p$-singular elements in Chevalley groups in characteristic $p$, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 169–182. MR 1829478
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
Daniele Garzoni and Eilidh McKemmie, On the probability of generating invariably a finite simple group, J. Pure Appl. Algebra 227 (2023), no. 6, Paper No. 107284, 37. MR 4521747, DOI 10.1016/j.jpaa.2022.107284
Ofir Gorodetsky, A polynomial analogue of Landau’s theorem and related problems, Mathematika 63 (2017), no. 2, 622–665. MR 3706601, DOI 10.1112/S0025579317000092
Robert M. Guralnick, Conjugacy classes of derangements in finite transitive groups, Tr. Mat. Inst. Steklova 292 (2016), no. Algebra, Geometriya i Teoriya Chisel, 118–123; English transl., Proc. Steklov Inst. Math. 292 (2016), no. 1, 112–117. MR 3628456, DOI 10.1134/S0371968516010076
Scott Harper, Shintani descent, simple groups and spread, J. Algebra 578 (2021), 319–355. MR 4234804, DOI 10.1016/j.jalgebra.2021.02.021
Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
Tomasz Łuczak and László Pyber, On random generation of the symmetric group, Combin. Probab. Comput. 2 (1993), no. 4, 505–512. MR 1264722, DOI 10.1017/S0963548300000869
Martin W. Liebeck, Jan Saxl, and Gary M. Seitz, Subgroups of maximal rank in finite exceptional groups of Lie type, Proc. London Math. Soc. (3) 65 (1992), no. 2, 297–325. MR 1168190, DOI 10.1112/plms/s3-65.2.297
Patrick Meisner, Erdős’ multiplication table problem for function fields and symmetric groups, arXiv:1804.08483v2 (2018).
Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737, DOI 10.1017/CBO9780511994777
G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299, DOI 10.1017/CBO9780511526015
G. M. Seitz, Generation of finite groups of Lie type, Trans. Amer. Math. Soc. 271 (1982), no. 2, 351–407., DOI 10.1090/S0002-9947-1982-0654839-1
J.P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. 40 (2003), no. 4, 429–440., DOI 10.1090/S0273-0979-03-00992-3

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Conjugacy classes of derangements in finite groups of Lie type
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Abstract: