Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree

Liguo He; Yaping Liu; Jianwei Lu

doi:doi:10.11648/j.mcs.20160101.14

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Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree

Liguo He, Yaping Liu, Jianwei Lu

Published in Mathematics and Computer Science (Volume 1, Issue 1)

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), 2024. 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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TY  - JOUR
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
PY  - 2016
N1  - https://doi.org/10.11648/j.mcs.20160101.14
DO  - 10.11648/j.mcs.20160101.14
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 17
EP  - 20
PB  - Science Publishing Group
SN  - 2575-6028
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

Liguo He

Dept. of Math., Shenyang University of Technology, Shenyang, PR China
Yaping Liu

Dept. of Math., Shenyang University of Technology, Shenyang, PR China
Jianwei Lu

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

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
PY  - 2016
N1  - https://doi.org/10.11648/j.mcs.20160101.14
DO  - 10.11648/j.mcs.20160101.14
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 17
EP  - 20
PB  - Science Publishing Group
SN  - 2575-6028
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  -

Copy | Download