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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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1435-5337
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Volume 29, Issue 2

Issues

Finite groups have more conjugacy classes

Barbara Baumeister / Attila Maróti / Hung P. Tong-Viet
Published Online: 2016-05-27 | DOI: https://doi.org/10.1515/forum-2015-0090

Abstract

We prove that for every ϵ>0 there exists a δ>0 such that every group of order n3 has at least δlog2n/(log2log2n)3+ϵ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order n has more than log3n conjugacy classes. We answer Bertram’s question in the affirmative for groups with a trivial solvable radical.

Keywords: Finite groups; number of conjugacy classes; simple groups

MSC 2010: 20E45; 20D06; 20P99

References

  • [1]

    Babai L. and Pyber L., Permutation groups without exponentially many orbits on the power set, J. Combin. Theory Ser. A 66 (1994), no. 1, 160–168. Google Scholar

  • [2]

    Bertram E. A., Lower bounds for the number of conjugacy classes in finite groups, Ischia Group Theory 2004, Contemp. Math. 402, American Mathematical Society, Providence (2006), 95–117. Google Scholar

  • [3]

    Bertram E. A., New reductions and logarithmic lower bounds for the number of conjugacy classes in finite groups., Bull. Aust. Math. Soc. 87 (2013), no. 3, 406–424. Google Scholar

  • [4]

    Bosma W., Cannon J. and Playoust C., The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265. Google Scholar

  • [5]

    Brauer R., Representations of finite groups, Lect. Modern Math. 1 (1963), 133–175. Google Scholar

  • [6]

    Carter R. W., Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Pure Appl. Math., John Wiley & Sons, New York, 1985. Google Scholar

  • [7]

    Cartwright M., The number of conjugacy classes of certain finite groups, Q. J. Math. Oxf. II. Ser. 36 (1985), no. 144, 393–404. Google Scholar

  • [8]

    Conway J. H., Curtis R. T., Norton S. P., Parker R. A. and Wilson R. A., An ATLAS of Finite Groups, Clarendon Press, Oxford, 1985. Google Scholar

  • [9]

    Dornhoff L., Group Representation Theory. Part A: Ordinary Representation Theory, Pure Appl. Math. 7., Marcel Dekker, New York, 1971. Google Scholar

  • [10]

    Erdős P. and Turán P., On some problems of a statistical group-theory. IV, Acta Math. Acad. Sci. Hung. 19 (1968), 413–435. Google Scholar

  • [11]

    Foulser D. A., Solvable primitive permutation groups of low rank, Trans. Amer. Math. Soc. 143 (1969), 1–54. Google Scholar

  • [12]

    Fulman J. and Guralnick R., 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. Google Scholar

  • [13]

    Jaikin-Zapirain A., On the number of conjugacy classes of finite nilpotent groups, Adv. Math. 227 (2011), no. 3, 1129–1143. Google Scholar

  • [14]

    Keller T. M., Finite groups have even more conjugacy classes, Israel J. Math. 181 (2011), 433–444. Google Scholar

  • [15]

    Landau E., Über die Klassenzahl der binären quadratischen Formen von negativer Discriminante, Math. Ann. 56 (1903), no. 4, 671–676. Google Scholar

  • [16]

    Lübeck F., Character degrees and their multiplicities for some groups of Lie type of rank <9, www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html.

  • [17]

    Malle G., Extensions of unipotent characters and the inductive McKay condition, J. Algebra 320 (2008), no. 7, 2963–2980. Google Scholar

  • [18]

    Maróti A., On elementary lower bounds for the partition function, Integers 3 (2003), Paper No. A10. Google Scholar

  • [19]

    Newman M., A bound for the number of conjugacy classes in a group, J. Lond. Math. Soc. 43 (1968), 108–110. Google Scholar

  • [20]

    Pyber L., Finite groups have many conjugacy classes, J. Lond. Math. Soc. (2) 46 (1992), no. 2, 239–249. Google Scholar

  • [21]

    The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.7.5, 2014, www.gap-system.org.

About the article


Received: 2015-05-13

Revised: 2016-02-16

Published Online: 2016-05-27

Published in Print: 2017-03-01


The second author was supported by an Alexander von Humboldt Fellowship for Experienced Researchers, by MTA Rényi “Lendület” Groups and Graphs Research Group, and by OTKA grants K84233 and K115799. Tong-Viet’s work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 93408). Part of the work was done while the last author held a position at the CRC 701 within the project C13 ‘The geometry and combinatorics of groups’. The first and second authors also wish to thank the CRC 701 for its support.


Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 259–275, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0090.

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