Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements
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- by Jason Fulman and Robert Guralnick PDF
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Abstract:
We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).References
- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
- George E. Andrews, Partitions, $q$-series and the Lusztig-Macdonald-Wall conjectures, Invent. Math. 41 (1977), no. 1, 91–102. MR 446991, DOI 10.1007/BF01390165
- 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
- M. Aschbacher and R. Guralnick, Solvable generation of groups and Sylow subgroups of the lower central series, J. Algebra 77 (1982), no. 1, 189–201. MR 665173, DOI 10.1016/0021-8693(82)90286-1
- David Benson, Walter Feit, and Roger Howe, Finite linear groups, the Commodore 64, Euler and Sylvester, Amer. Math. Monthly 93 (1986), no. 9, 717–719. MR 863974, DOI 10.2307/2322289
- Nigel Boston, Walter Dabrowski, Tuval Foguel et al., The proportion of fixed-point-free elements of a transitive permutation group, Comm. Algebra 21 (1993), no. 9, 3259–3275. MR 1228762, DOI 10.1080/00927879308824728
- John R. Britnell and Mark Wildon, On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension, Bull. Lond. Math. Soc. 40 (2008), no. 5, 897–906. MR 2439655, DOI 10.1112/blms/bdn073
- 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
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- Persi Diaconis, Jason Fulman, and Robert Guralnick, On fixed points of permutations, J. Algebraic Combin. 28 (2008), no. 1, 189–218. MR 2420785, DOI 10.1007/s10801-008-0135-2
- John D. Dixon, Random sets which invariably generate the symmetric group, Discrete Math. 105 (1992), no. 1-3, 25–39. MR 1180190, DOI 10.1016/0012-365X(92)90129-4
- Walter Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. J. 27 (1960), 91–94. MR 109810
- Feller, W., An introduction to probability theory and its applications, second edition, John Wiley & Sons, 1957.
- Michael D. Fried, Robert Guralnick, and Jan Saxl, Schur covers and Carlitz’s conjecture, Israel J. Math. 82 (1993), no. 1-3, 157–225. MR 1239049, DOI 10.1007/BF02808112
- Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51–85. MR 1864086, DOI 10.1090/S0273-0979-01-00920-X
- 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
- 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
- Fulman, J. and Guralnick, R., Derangements in finite classical groups for actions related to extension field and imprimitive subgroups, preprint.
- Fulman, J. and Guralnick, R., Derangements in finite classical groups for subspace actions, preprint.
- Fulman, J. and Guralnick, R., The probability of generating an irreducible subgroup, preprint.
- Fulman, J. and Guralnick, R., Conjugacy classes in orthogonal and symplectic groups, preprint.
- Jason Fulman, Peter M. Neumann, and Cheryl E. Praeger, A generating function approach to the enumeration of matrices in classical groups over finite fields, Mem. Amer. Math. Soc. 176 (2005), no. 830, vi+90. MR 2145026, DOI 10.1090/memo/0830
- Patrick X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175–179. MR 276318, DOI 10.1007/BF01113339
- David Gluck, Characters and random walks on finite classical groups, Adv. Math. 129 (1997), no. 1, 46–72. MR 1458412, DOI 10.1006/aima.1996.1635
- 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
- Rod Gow and C. Ryan Vinroot, Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents, J. Group Theory 11 (2008), no. 3, 299–331. MR 2419003, DOI 10.1515/JGT.2008.017
- Robert M. Guralnick, Intersections of conjugacy classes and subgroups of algebraic groups, Proc. Amer. Math. Soc. 135 (2007), no. 3, 689–693. MR 2262864, DOI 10.1090/S0002-9939-06-08544-3
- Robert M. Guralnick, William M. Kantor, and Jan Saxl, The probability of generating a classical group, Comm. Algebra 22 (1994), no. 4, 1395–1402. MR 1261266, DOI 10.1080/00927879408824912
- Robert M. Guralnick, Martin W. Liebeck, Dugald Macpherson, and Gary M. Seitz, Modules for algebraic groups with finitely many orbits on subspaces, J. Algebra 196 (1997), no. 1, 211–250. MR 1474171, DOI 10.1006/jabr.1997.7068
- Robert M. Guralnick, Peter Müller, and Jan Saxl, The rational function analogue of a question of Schur and exceptionality of permutation representations, Mem. Amer. Math. Soc. 162 (2003), no. 773, viii+79. MR 1955160, DOI 10.1090/memo/0773
- Robert M. Guralnick and Geoffrey R. Robinson, On the commuting probability in finite groups, J. Algebra 300 (2006), no. 2, 509–528. MR 2228209, DOI 10.1016/j.jalgebra.2005.09.044
- Robert M. Guralnick and Pham Huu Tiep, The non-coprime $k(GV)$ problem, J. Algebra 293 (2005), no. 1, 185–242. MR 2173972, DOI 10.1016/j.jalgebra.2005.02.001
- Robert M. Guralnick and Pham Huu Tiep, Symmetric powers and a problem of Kollár and Larsen, Invent. Math. 174 (2008), no. 3, 505–554. MR 2453600, DOI 10.1007/s00222-008-0140-z
- Robert Guralnick and Daqing Wan, Bounds for fixed point free elements in a transitive group and applications to curves over finite fields, Israel J. Math. 101 (1997), 255–287. MR 1484879, DOI 10.1007/BF02760932
- James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976, DOI 10.1090/surv/043
- I. M. Isaacs, Characters of $\pi$-separable groups, J. Algebra 86 (1984), no. 1, 98–128. MR 727371, DOI 10.1016/0021-8693(84)90058-9
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Noriaki Kawanaka, On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan 29 (1977), no. 3, 425–450. MR 450383, DOI 10.2969/jmsj/02930425
- 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
- L. G. Kovács and Geoffrey R. Robinson, On the number of conjugacy classes of a finite group, J. Algebra 160 (1993), no. 2, 441–460. MR 1244923, DOI 10.1006/jabr.1993.1196
- Martin W. Liebeck, Benjamin M. S. Martin, and Aner Shalev, On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function, Duke Math. J. 128 (2005), no. 3, 541–557. MR 2145743, DOI 10.1215/S0012-7094-04-12834-9
- Martin W. Liebeck, E. A. O’Brien, Aner Shalev, and Pham Huu Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939–1008. MR 2654085, DOI 10.4171/JEMS/220
- Martin W. Liebeck and László Pyber, Upper bounds for the number of conjugacy classes of a finite group, J. Algebra 198 (1997), no. 2, 538–562. MR 1489911, DOI 10.1006/jabr.1997.7158
- 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, DOI 10.1016/j.jalgebra.2006.10.047
- Liebeck, M. and Seitz, G., Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, preprint.
- Martin W. Liebeck and Aner Shalev, Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), no. 2, 317–367. MR 2116277, DOI 10.1007/s00222-004-0390-3
- Martin W. Liebeck and Aner Shalev, Character degrees and random walks in finite groups of Lie type, Proc. London Math. Soc. (3) 90 (2005), no. 1, 61–86. MR 2107038, DOI 10.1112/S0024611504014935
- Martin W. Liebeck and Aner Shalev, Maximal subgroups of symmetric groups, J. Combin. Theory Ser. A 75 (1996), no. 2, 341–352. MR 1401008, DOI 10.1006/jcta.1996.0082
- Frank Lübeck, Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135–169. MR 1901354, DOI 10.1112/S1461157000000838
- Lübeck, F., http://www.math.rwth-aachen.de/$\sim {}$Frank.Luebeck/.
- 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
- G. Lusztig, On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), no. 3, 201–213. MR 419635, DOI 10.1007/BF01403067
- G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125–175. MR 463275, DOI 10.1007/BF01390002
- I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bull. Austral. Math. Soc. 23 (1981), no. 1, 23–48. MR 615131, DOI 10.1017/S0004972700006882
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Attila Maróti, Bounding the number of conjugacy classes of a permutation group, J. Group Theory 8 (2005), no. 3, 273–289. MR 2137971, DOI 10.1515/jgth.2005.8.3.273
- David K. Maslen and Daniel N. Rockmore, Separation of variables and the computation of Fourier transforms on finite groups. I, J. Amer. Math. Soc. 10 (1997), no. 1, 169–214. MR 1396896, DOI 10.1090/S0894-0347-97-00219-1
- Peter M. Neumann and Cheryl E. Praeger, Cyclic matrices over finite fields, J. London Math. Soc. (2) 52 (1995), no. 2, 263–284. MR 1356142, DOI 10.1112/jlms/52.2.263
- A. M. Odlyzko, Asymptotic enumeration methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1063–1229. MR 1373678
- Geoffrey R. Robinson, Bounding numbers and heights of characters in $p$-constrained groups, Finite groups 2003, Walter de Gruyter, Berlin, 2004, pp. 307–317. MR 2125082
- Gary M. Seitz, Generation of finite groups of Lie type, Trans. Amer. Math. Soc. 271 (1982), no. 2, 351–407. MR 654839, DOI 10.1090/S0002-9947-1982-0654839-1
- Jean-Pierre Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 429–440. MR 1997347, DOI 10.1090/S0273-0979-03-00992-3
- Aner Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math. (2) 170 (2009), no. 3, 1383–1416. MR 2600876, DOI 10.4007/annals.2009.170.1383
- Takuro Shintani, Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Japan 28 (1976), no. 2, 396–414. MR 414730, DOI 10.2969/jmsj/02820396
- Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- Pham Huu Tiep, Finite groups admitting Grassmannian 4-designs, J. Algebra 306 (2006), no. 1, 227–243. MR 2271581, DOI 10.1016/j.jalgebra.2006.01.056
- J. H. van Lint and R. M. Wilson, A course in combinatorics, Cambridge University Press, Cambridge, 1992. MR 1207813
- G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc. 3 (1963), 1–62. MR 0150210
- G. E. Wall, Some applications of the Eulerian functions of a finite group, J. Austral. Math. Soc. 2 (1961/1962), 35–59. MR 0125156
Additional Information
- Jason Fulman
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- MR Author ID: 332245
- Email: fulman@usc.edu
- Robert Guralnick
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- MR Author ID: 78455
- Email: guralnic@usc.edu
- Received by editor(s): October 16, 2009
- Received by editor(s) in revised form: July 21, 2010
- Published electronically: February 7, 2012
- Additional Notes: The first author was partially supported by National Science Foundation grants DMS 0503901, DMS 0802082, and National Security Agency grants MDA904-03-1-004, H98230-08-1-0133.
The second author was partially supported by National Science Foundation grants DMS 0140578 and DMS 0653873 - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3023-3070
- MSC (2010): Primary 20G40, 20B15
- DOI: https://doi.org/10.1090/S0002-9947-2012-05427-4
- MathSciNet review: 2888238