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Analysis on the Properties of a Permutation Group

Received: 17 October 2016     Accepted: 21 November 2016     Published: 27 December 2016
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Abstract

The structures of the subgroups play an important role in the study of the nature of symmetric groups. We calculate the 11300 subgroups of the permutation group S7 by group-theoretical approach. The analytic expressions for the numbers of subgroups are obtained. The subgroups of the permutation group S7 are all represented in an alternative way for further analysis and applications.

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 1)
DOI 10.11648/j.ijtam.20170301.13
Page(s) 19-24
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

Permutation Group, Subgroup, Lagrange’s Theorem, Cayley’s Theorem

References
[1] A.-S. Elsenhans, Improved methods for the construction of relative invariants for permutation groups, J. Sym. Comp., 79 (2016) 211.
[2] G.-D. Deng, Y. Fan, Permutation-like matrix groups with a maximal cycle of power of odd prime length, Linear Algebra Appl., 480 (2015) 1.
[3] C.-H. Li, C. E. Praeger, On finite permutation groups with a transitive cyclic subgroup, J. Algebra, 349 (2012) 117.
[4] J.-J. Cannon, B.-C. Cox and D.-F. Holt, Computing the subgroups of a permutation group, J. Sym. Comp. 31 (2001) 149.
[5] B.-W. Huang, X.-J. Liao, Y.-X. Lu and X.-H. Wang, The subgroups of symmetric group S7, J. Wuhan Univ. (Nat. Sci. Ed), 51 (2005) 39.
[6] J.-S. Wang, M.-Z. Lin, Solutions to the Permutation Group with Mathematica, J. Hanshan Normal Univ., 29 (2008) 14.
[7] W.-R. Unger, Computing the soluble radical of a permutation group, J. Algebra, 300 (2006) 305.
[8] J. Gallian, Contemporary Abstract Algebra (6th ed.), (Boston: Houghton Mifflin, 2006).
[9] R. R. Roth, A History of Lagrange’s Theorem on Groups, Math. Mag. 74 (2001) 99.
[10] H. Kurzweil and B. Stellmacher, The theory of Finite Groups (Springer-Verlag, New York, 2004).
[11] Z.-Q. Ma, Group Theory for Physicists, (World Scientific, Singapore, 2007).
[12] A. Cayley Esq., LXV. On the theory of groups as depending on the symbolic equation n=1.-Part II, Phil. Magazine Ser. 4, 7 (1854) 408.
[13] E. C. Nummela, Cayley’s Theorem for Topological Groups, Am. Math. Mon., 87 (1980) 202.
[14] L. N. Childs, J. Corradino, Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups, J. Algebra, 308 (2007) 236.
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  • APA Style

    Xiao-Yan Gu, Jian-Qiang Sun. (2016). Analysis on the Properties of a Permutation Group. International Journal of Theoretical and Applied Mathematics, 3(1), 19-24. https://doi.org/10.11648/j.ijtam.20170301.13

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    ACS Style

    Xiao-Yan Gu; Jian-Qiang Sun. Analysis on the Properties of a Permutation Group. Int. J. Theor. Appl. Math. 2016, 3(1), 19-24. doi: 10.11648/j.ijtam.20170301.13

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    AMA Style

    Xiao-Yan Gu, Jian-Qiang Sun. Analysis on the Properties of a Permutation Group. Int J Theor Appl Math. 2016;3(1):19-24. doi: 10.11648/j.ijtam.20170301.13

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  • @article{10.11648/j.ijtam.20170301.13,
      author = {Xiao-Yan Gu and Jian-Qiang Sun},
      title = {Analysis on the Properties of a Permutation Group},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {3},
      number = {1},
      pages = {19-24},
      doi = {10.11648/j.ijtam.20170301.13},
      url = {https://doi.org/10.11648/j.ijtam.20170301.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170301.13},
      abstract = {The structures of the subgroups play an important role in the study of the nature of symmetric groups. We calculate the 11300 subgroups of the permutation group S7 by group-theoretical approach. The analytic expressions for the numbers of subgroups are obtained. The subgroups of the permutation group S7 are all represented in an alternative way for further analysis and applications.},
     year = {2016}
    }
    

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    AU  - Jian-Qiang Sun
    Y1  - 2016/12/27
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    DO  - 10.11648/j.ijtam.20170301.13
    T2  - International Journal of Theoretical and Applied Mathematics
    JF  - International Journal of Theoretical and Applied Mathematics
    JO  - International Journal of Theoretical and Applied Mathematics
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    AB  - The structures of the subgroups play an important role in the study of the nature of symmetric groups. We calculate the 11300 subgroups of the permutation group S7 by group-theoretical approach. The analytic expressions for the numbers of subgroups are obtained. The subgroups of the permutation group S7 are all represented in an alternative way for further analysis and applications.
    VL  - 3
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Author Information
  • Physics Department, East China University of Science and Technology, Shanghai, China

  • College of Information Science and Technology, Hainan University, Haikou, China

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