Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements

Transactions of the American Mathematical Society

Volumn 364, Issue 6, 2012, Pages 3023-3070

  Fulman, Jason ^a   Guralnick, Robert ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Chevalley groups; Derangements; Generating function; Number of conjugacy classes; Partition; Simple group

Indexed keywords

EID: 84857674126     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-2012-05427-4     Document Type: Article

References (70)

1. Andrews, G., The theory of partitions, Addison-Wesley, Reading, Mass., 1976. MR0557013 (58:27738)
2. Andrews, G., Partitions, q-series and the Lusztig-Macdonald-Wall conjectures, Invent. Math. 41 (1977), 91-102. MR0446991 (56:5307)
3. Aschbacher, M., On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), 469-514. MR746539 (86a:20054)
4. Aschbacher, M. and Guralnick, R., Solvable generation of groups and Sylow subgroups of the lower central series, J. Algebra 77 (1982), 189-201. MR665173 (84c:20025)
5. Benson, D., Feit, W., and Howe, R., Finite linear groups, the Commodore 64, Euler and Sylvester, Amer. Math. Monthly 93 (1986), 717-719. MR863974 (87m:05011)
6. Boston, N., Dabrowski, W., Foguel, T., et al., The proportion of fixed point free elements of a transitive permutation group, Comm. Algebra 21 (1993), 3259-3275. MR1228762 (94e:20002)
7. Britnell, J. and Wildon, M., On the distribution of conjugacy classes between cosets of a finite group, Bull. London Math. Soc. 40 (2008), 897-906. MR2439655 (2009g:20057)
8. Cameron, P. and Cohen, A., On the number of fixed point free elements in a permutation group, Discrete Math. 106/107 (1992), 135-138. MR1181907 (93f:20004)
9. Carter, R., Finite groups of Lie type: Conjugacy classes and complex characters, John Wiley & Sons, New York, 1985. MR794307 (87d:20060)
10. Diaconis, P., Fulman, J., and Guralnick, R., On fixed points of permutations, J. Algebraic Combin. 28 (2008), 189-218. MR2420785 (2009g:20002)
11. Dixon, J., Random sets which invariably generate the symmetric group, Discrete Math. 105 (1992), 25-39. MR1180190 (93e:20004)
12. Feit, W. and Fine, N. J., Pairs of commuting matrices over a finite field, Duke Math. J. 27 (1960), 91-94. MR0109810 (22:695)
13. Feller, W., An introduction to probability theory and its applications, second edition, John Wiley & Sons, 1957.
14. Fried, M., Guralnick, R. and Saxl, J., Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157-225. MR1239049 (94j:12007)
15. Fulman, J., Random matrix theory over finite fields, Bull. Amer. Math. Soc. 39 (2002), 51-85. MR1864086 (2002i:60012)
16. Fulman, J., Cycle indices for the finite classical groups, J. Group Theory 2 (1999), 251-289. MR1696313 (2001d:20045)
17. Fulman, J. and Guralnick, R., Derangements in simple and primitive groups. In: Groups, Combinatorics, and Geometry (Durham 2001), A. Ivanov et al., Eds., World Scientific, 2003. MR1994962 (2004e:20003)
18. Fulman, J. and Guralnick, R., Derangements in finite classical groups for actions related to extension field and imprimitive subgroups, preprint.
19. Fulman, J. and Guralnick, R., Derangements in finite classical groups for subspace actions, preprint.
20. Fulman, J. and Guralnick, R., The probability of generating an irreducible subgroup, preprint.
21. Fulman, J. and Guralnick, R., Conjugacy classes in orthogonal and symplectic groups, preprint.
22. Fulman, J., Neumann, P.M. and Praeger, C.E., A generating function approach to the enumeration of matrices in finite classical groups, Mem. Amer. Math. Soc. 176 (2005), no. 830. MR2145026 (2006b:05125)
23. Gallagher, P., The number of conjugacy classes of a finite group, Math. Z. 118 (1970), 175-179. MR0276318 (43:2065)
24. Gluck, D., Characters and random walks on finite classical groups, Adv. Math. 129 (1997), 46-72. MR1458412 (98g:20027)
25. Gorenstein, D., Lyons, R. and Solomon, R., The classification of the finite simple groups. Number 3, American Mathematical Society, Providence, RI, 1998. MR1490581 (98j:20011)
26. Gow, R. and Vinroot, C. R., Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents, J. Group Theory 11 (2008), 299-331. MR2419003 (2009c:20082)
27. Guralnick, R., Intersections of conjugacy classes and subgroups of algebraic groups, Proc. Amer. Math. Soc. 135 (2007), 689-693. MR2262864 (2007i:20071)
28. Guralnick, R., Kantor, W. and Saxl, J., The probability of generating a classical group, Comm. Algebra 22 (1994), 1395-1402. MR1261266 (95a:20030)
29. Guralnick, R, Liebeck, M., Macpherson, D. and Seitz, G., Modules for algebraic groups with finitely many orbits on subspaces, J. Algebra 196 (1997), 211-250. MR1474171 (98j:20058)
30. Guralnick, R., Müller, P. and Saxl, J., The rational function analogue of a question of Schur and exceptionality of permutation representations, Mem. Amer. Math. Soc. 162 (2003), no. 773. MR1955160 (2004d:12005)
31. Guralnick, R. and Robinson, G., On the commuting probability in finite groups, J. Algebra 300 (2006), 509-528. MR2228209 (2007g:60011)
32. Guralnick, R. and Tiep, P., The non-coprime k(GV) problem, J. Algebra 293 (2005), 185-242. MR2173972 (2006g:20018)
33. Guralnick, R. and Tiep, P., Symmetric powers and a problem of Kollár and Larsen, Invent. Math. 174 (2008), 505-554. MR2453600 (2009j:20065)
34. Guralnick, R. and Wan, D., Bounds for fixed point free elements in a transitive group and applications to curves over finite fields, Israel J. Math. 101 (1997), 255-287. MR1484879 (98j:12002)
35. Humphreys, J., Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43, American Math. Society, Providence, 1995. MR1343976 (97i:20057)
36. Isaacs, I. M., Characters of π-separable groups, J. Algebra 86 (1984), 98-128. MR727371 (85h:20012)
37. Jantzen, J., Representations of algebraic groups, second edition, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003. MR2015057 (2004h:20061)
38. Kawanaka, N., On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan 29 (1977), 425-449. MR0450383 (56:8678)
39. Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, 129, Cambridge University Press, Cambridge, 1990. MR1057341 (91g:20001)
40. Kovács, L. and Robinson, G., On the number of conjugacy classes of a finite group, J. Algebra 160 (1993), 441-460. MR1244923 (94m:20057)
41. Liebeck, M., Martin B., and Shalev, A., On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function. Duke Math. J. 128 (2005), 541-557. MR2145743 (2006b:20043)
42. Liebeck, M., O'Brien, E., Shalev, A., and Tiep, P., On the Ore conjecture, J. European Math. Soc. 12 (2010), 939-1008. MR2654085 (2011e:20016)
43. Liebeck, M. and Pyber, L., Upper bounds for the number of conjugacy classes of a finite group, J. Algebra 198 (1997), 538-562. MR1489911 (99c:20023)
44. Liebeck, M., Pyber, L. and Shalev, A., On a conjecture of G. E. Wall, J. Algebra 317 (2007), 184-197. MR2360145 (2009b:20048)
45. Liebeck, M. and Seitz, G., Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, preprint.
46. Liebeck, M. and Shalev, A., Fuchsian groups, finite simple groups and representation varieties, Invent. Math. 159 (2005), 317-367. MR2116277 (2005j:20065)
47. Liebeck, M. and Shalev, A, Character degrees and random walks in finite groups of Lie type, Proc. London Math. Soc. 90 (2005), 61-86. MR2107038 (2006h:20016)
48. Liebeck, M. and Shalev, A., Maximal subgroups of symmetric groups, J. Combin. Theory Ser. A 75 (1996), 341-352. MR1401008 (98b:20005)
49. Lübeck, F., Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math. 4 (2001), 135-169. MR1901354 (2003e:20013)
50. Lübeck, F., http://www.math.rwth-aachen.de/~ Frank.Luebeck/.
51. Luczak, T. and Pyber, L., On random generation of the symmetric group, Combin. Probab. Comput. 2 (1993), 505-512. MR1264722 (95b:20004)
52. Lusztig, G., On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), 201-213. MR0419635 (54:7653)
53. Lusztig, G., Irreducible representations of the finite classical groups, Invent. Math 43 (1977), 125-176. MR0463275 (57:3228)
54. Macdonald, I., Numbers of conjugacy classes in some finite classical groups, Bull. Austral. Math. Soc. 23 (1981), 23-48. MR615131 (82j:20092)
55. Macdonald, I., Symmetric functions and Hall polynomials, second edition, Clarendon Press, Oxford, 1995. MR1354144 (96h:05207)
56. Maróti, A., Bounding the number of conjugacy classes in a permutation group, J. Group Theory 8 (2005), 273-289. MR2137971 (2006e:20054)
57. Maslen, D. and Rockmore, D., Separation of variables and the computation of Fourier transforms on finite groups. I, J. Amer. Math. Soc. 10 (1997), 169-214. MR1396896 (97i:20019)
58. Neumann, P.M. and Praeger, C.E., Cyclic matrices over finite fields, J. London Math. Soc. 52 (1995), 263-284. MR1356142 (96j:15017)
59. Odlyzko, A.M., Asymptotic enumeration methods, Chapter 22 in Handbook of Combinatorics, Volume 2. MIT Press and Elsevier, 1995. MR1373678 (97b:05012)
60. Robinson, G., Bounding numbers and heights of characters in p-constrained groups. In: Finite Groups 2003, 307-317, Walter de Gruyter GmbH & Co. KG, Berlin, 2004. MR2125082 (2006e:20017)
61. Seitz, G., Generation of finite groups of Lie type, Trans. Amer. Math. Soc. 271 (1982), 351-407. MR654839 (83h:20021)
62. Serre, J.-P., On a theorem of Jordan, Bull. Amer. Math. Soc. 40 (2003), 429-440. MR1997347 (2004f:11040)
63. Shalev, A., Word maps, conjugacy classes, and a non-commutativeWaring-type theorem, Annals Math. 170 (2009), 1383-1416. MR2600876
64. Shintani, T., Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Japan 28 (1976), 396-414. MR0414730 (54:2825)
65. Steinberg, R., Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., No. 80, 1968. MR0230728 (37:6288)
66. Steinberg, R., Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. MR0466335 (57:6215)
67. Tiep, P., Finite groups admitting grassmannian 4-designs, J. Algebra 306 (2006), 227-243. MR2271581 (2007j:20008)
68. van Lint, J.H. and Wilson, R.M., A Course in Combinatorics, Cambridge University Press, Cambridge, England, 1992. MR1207813 (94g:05003)
69. Wall, G. E., On the conjugacy classes in the unitary, symplectic, and orthogonal groups, J. Aust. Math. Soc. 3 (1963), 1-63. MR0150210 (27:212)
70. Wall, G. E., Some applications of the Eulerian functions of a finite group, J. Aust. Math. Soc. 2 (1961), 35-59. MR0125156 (23:A2461)