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Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree

Received: 11 April 2016     Accepted: 3 May 2016     Published: 28 May 2016
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Abstract

For a finite group G, we write to denote the prime divisor set of the various conjugacy class lengths of G and the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that can be bounded by . Let G be an almost simple group G such that the graph built on element orders is a tree. By using Lucido’s classification theorem, we prove except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with a forest, the inequality is true.

Published in Mathematics and Computer Science (Volume 1, Issue 1)
DOI 10.11648/j.mcs.20160101.14
Page(s) 17-20
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Prime Graph, Conjugacy Class Length, Almost Simple Group

References
[1] C. Casolo, Prime divisors of conjugacy class lengths in finite groups, Rend. Mat. Acc. Lincei, 1991, Ser. 9, 2: 111-113.
[2] C. Casolo, Finite groups with small conjugacy classes, Manuscr. Math., 1994, 82: 171-189.
[3] D. Chillig, M. Herzog, On the length of conjugacy classes of finite groups, J. Algebra, 1990, 131: 110-125.
[4] L. Dornhoff, Group representation theory, Part A: Ordinary representation theory, Marcel Dekker, New York, 1971.
[5] P. Ferguson, Connections between prime divisors of the conjugacy classes and prime divisors of , J. Algebra, 1991, 143: 25-28.
[6] The GAP Group, GAP- Groups, algorithms, and programming, version 4.6,http://www.gap-system.org, 2013.
[7] D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc. 276, 1983 (vol.42).
[8] L. He, Y. Dong, Conjugacy class lengths of finite groups with disconnected prime graph, Int. J. Algebra, 2015, 9(5): 239 - 243.
[9] B. Huppert, Character theory of finite groups, DeGruyter Expositions in Mathematics 25, Walter de Gruyter & Co.: Berlin. New York, 1998.
[10] M. L. Lewis and D. L. White, Nonsolvable groups with no prime dividing three character degrees, J. Algebra, 2011, 336: 158-183.
[11] M. S. Lucido, Groups in which the prime graph is a tree, Bollettino U.M.I., 2002, Ser. 8, 5-B: 131-148.
[12] M. Suzuki, On a class of doubly transitive groups, Ann. Math., 1962, 75:105-145.
[13] J. P. Zhang, On the lengths of conjugacy classes, Comm. Algebra, 1998, 26(8): 2395-2400.
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  • APA Style

    Liguo He, Yaping Liu, Jianwei Lu. (2016). Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree. Mathematics and Computer Science, 1(1), 17-20. https://doi.org/10.11648/j.mcs.20160101.14

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    ACS Style

    Liguo He; Yaping Liu; Jianwei Lu. Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree. Math. Comput. Sci. 2016, 1(1), 17-20. doi: 10.11648/j.mcs.20160101.14

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    AMA Style

    Liguo He, Yaping Liu, Jianwei Lu. Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree. Math Comput Sci. 2016;1(1):17-20. doi: 10.11648/j.mcs.20160101.14

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  • @article{10.11648/j.mcs.20160101.14,
      author = {Liguo He and Yaping Liu and Jianwei Lu},
      title = {Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree},
      journal = {Mathematics and Computer Science},
      volume = {1},
      number = {1},
      pages = {17-20},
      doi = {10.11648/j.mcs.20160101.14},
      url = {https://doi.org/10.11648/j.mcs.20160101.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20160101.14},
      abstract = {For a finite group G, we write  to denote the prime divisor set of the various conjugacy class lengths of G and  the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that  can be bounded by . Let G be an almost simple group G such that the graph  built on element orders is a tree. By using Lucido’s classification theorem, we prove  except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with  a forest, the inequality  is true.},
     year = {2016}
    }
    

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    T1  - Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree
    AU  - Liguo He
    AU  - Yaping Liu
    AU  - Jianwei Lu
    Y1  - 2016/05/28
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    DO  - 10.11648/j.mcs.20160101.14
    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
    JO  - Mathematics and Computer Science
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    EP  - 20
    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.mcs.20160101.14
    AB  - For a finite group G, we write  to denote the prime divisor set of the various conjugacy class lengths of G and  the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that  can be bounded by . Let G be an almost simple group G such that the graph  built on element orders is a tree. By using Lucido’s classification theorem, we prove  except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with  a forest, the inequality  is true.
    VL  - 1
    IS  - 1
    ER  - 

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Author Information
  • Dept. of Math., Shenyang University of Technology, Shenyang, PR China

  • Dept. of Math., Shenyang University of Technology, Shenyang, PR China

  • Dept. of Math., Shenyang University of Technology, Shenyang, PR China

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