10.1007/s40096-020-00361-6

Some numerical results on the wreath product of Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_p$$\end{document} and Zpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{p^n}$$\end{document}

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  • Mansour Hashemi*1,
  • Mina Pirzadeh1,
  • Shoja Ali Gorjian2,
  1. Faculty of Mathematical Sciences, University of Guilan, Rasht, IR
  2. Department of Mathematics, University of Guilan, Rasht, IR

Published in Issue 2021-01-02

How to Cite

Hashemi, M., Pirzadeh, M., & Gorjian, S. A. (2021). Some numerical results on the wreath product of Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_p$$\end{document} and Zpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{p^n}$$\end{document}. Mathematical Sciences, 15(3 (September 2021). https://doi.org/10.1007/s40096-020-00361-6
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Abstract

Abstract For every positive integer n and for a prime number p , we denote the wreath product of Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_p$$\end{document} and Zpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{p^n}$$\end{document} by G ( n ,  p ). In this paper, we will consider three probabilistic concepts of finite groups. The first problem which we examine is the calculation of the k th -roots of elements in G ( n ,  p ) when k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document} . The second problem which is investigated is the computation of the k th -commutative degree of G ( n ,  p ) when k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 1$$\end{document} . In the end, for k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 1$$\end{document} we compute the probability that the commutator equation [xk,y]=g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[x^k,y]=g$$\end{document} has solution in G ( n ,  p ).

Keywords

  • Groups,
  • Wreath product,
  • kth-roots,
  • kth-commutativity degree

References

  1. Das (2005) On groups elements having square roots 31(2) (pp. 33-36)
  2. Doostie and Maghasedi (2009) Certain classes of groups with commutativity degree d(G)≤12documentclass[12pt]{minimal}
  3. usepackage{amsmath}
  4. usepackage{wasysym}
  5. usepackage{amsfonts}
  6. usepackage{amssymb}
  7. usepackage{amsbsy}
  8. usepackage{mathrsfs}
  9. usepackage{upgreek}
  10. setlength{oddsidemargin}{-69pt}
  11. begin{document}$$d(G)le frac{1}{2}$$end{document} (pp. 263-270)
  12. Erdos and Turan (1968) On some problems of a statistical group theory IV (pp. 413-435) https://doi.org/10.1007/BF01894517
  13. Guralnick (1982) Commutators and commutator subgroups 45(3) (pp. 319-330) https://doi.org/10.1016/S0001-8708(82)80008-X
  14. Gusafson (1973) What is the probability that two group elements commute? (pp. 1031-1034) https://doi.org/10.1080/00029890.1973.11993437
  15. Hashemi (2015) The commutativity degree of 2-generated groups of nilpotency class 2 (pp. 149-159)
  16. Hashemi and Pirzadeh (2019) The probability that the commutator equation [x,y]=gdocumentclass[12pt]{minimal}
  17. usepackage{amsmath}
  18. usepackage{wasysym}
  19. usepackage{amsfonts}
  20. usepackage{amssymb}
  21. usepackage{amsbsy}
  22. usepackage{mathrsfs}
  23. usepackage{upgreek}
  24. setlength{oddsidemargin}{-69pt}
  25. begin{document}$$[x, y]=g$$end{document} has solution in a finite group 7(2) (pp. 47-61)
  26. Hashemi and Polkouei (2017) Some numerical results on finite non-abelian 2documentclass[12pt]{minimal}
  27. usepackage{amsmath}
  28. usepackage{wasysym}
  29. usepackage{amsfonts}
  30. usepackage{amssymb}
  31. usepackage{amsbsy}
  32. usepackage{mathrsfs}
  33. usepackage{upgreek}
  34. setlength{oddsidemargin}{-69pt}
  35. begin{document}$$2$$end{document}-generator pdocumentclass[12pt]{minimal}
  36. usepackage{amsmath}
  37. usepackage{wasysym}
  38. usepackage{amsfonts}
  39. usepackage{amssymb}
  40. usepackage{amsbsy}
  41. usepackage{mathrsfs}
  42. usepackage{upgreek}
  43. setlength{oddsidemargin}{-69pt}
  44. begin{document}$$p$$end{document}-groups of nilpotency class two (pp. 589-599)
  45. Lucido and Pournaki (2005) Elements with square roots in finite groups 12(4) (pp. 677-690) https://doi.org/10.1142/S1005386705000647
  46. Mohd Ali and Sarmin (2010) On some problems in group theory of probabilistic nature 32(2) (pp. 35-41)
  47. Nath and Prajapati (2018) On the number of solutions of a generalized commutator equation in finite groups 156(1) (pp. 18-37) https://doi.org/10.1007/s10474-018-0863-2
  48. Pournakia and Sobha (2008) Probability that the commutator of two group elements is equal to a given elemente is equal to a given element 212(4) (pp. 727-734) https://doi.org/10.1016/j.jpaa.2007.06.013
  49. Sadeghieh and Doostie (2008) The nth-documentclass[12pt]{minimal}
  50. usepackage{amsmath}
  51. usepackage{wasysym}
  52. usepackage{amsfonts}
  53. usepackage{amssymb}
  54. usepackage{amsbsy}
  55. usepackage{mathrsfs}
  56. usepackage{upgreek}
  57. setlength{oddsidemargin}{-69pt}
  58. begin{document}$$n^{th}-$$end{document}roots of elements in finite groups 2(4) (pp. 347-356)
  59. Yahya (2012) The nth-documentclass[12pt]{minimal}
  60. usepackage{amsmath}
  61. usepackage{wasysym}
  62. usepackage{amsfonts}
  63. usepackage{amssymb}
  64. usepackage{amsbsy}
  65. usepackage{mathrsfs}
  66. usepackage{upgreek}
  67. setlength{oddsidemargin}{-69pt}
  68. begin{document}$$n^{th}-$$end{document} commutativity degree of some dihedral groups 34(2) (pp. 7-14)