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  • 1 Cairo University, Giza 12613, Egypt
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Abstract

Let G be a finite group and H a subgroup of G. We say that H is an -subgroup of G if NG (H) ∩ HgH for all gG; H is called weakly -embedded in G if G has a normal subgroup K such that HG = HK and HK is an -subgroup of G, where HG is the normal clousre of H in G, i. e., HG = 〈Hg|gG〉. In this paper, we study the p-nilpotence of a group G under the assumption that every subgroup of order d of a Sylow p-subgroup P of G with 1 < d < |P| is weakly -embedded in G. Many known results related to p-nilpotence of a group G are generalized.

  • [1]

    Asaad, M., Heliel, A. A. and Al-Shomranim, A. A., On weakly -subgroups of finite groups, Comm. Algebra, 40 (2012), 35403550.

  • [2]

    Asaad, M. and Ramadan, M., On weakly -embedded subgroups of finite groups, Comm. Algebra, 44 (2016), 45644574.

  • [3]

    Bianchi, M., Gillio Berta Mauri, A., Herzog, M. and Veradi, L., On finite solvable groups in which normality is a transitive relation, J. Group Theory, 3 (2000), 147156.

    • Search Google Scholar
    • Export Citation
  • [4]

    Chen, X., Guo, W. and Skiba, A. N., Some conditions under which a finite group belongs to a Baer-Local formation, Comm. Algebra, 42 (2014), 41884203.

    • Search Google Scholar
    • Export Citation
  • [5]

    Doerk, D. and Hawkes, T., Finite Solvable Groups, Walter de Gruyter (Berlin-New/York, 1992).

  • [6]

    Gorenstein, D., Finite Groups, Harper and Row (New York, 1968).

  • [7]

    Guo, X. and Wei, X., The influence of -subgroups on the structure of finite groups, J. Group Theory, 13 (2010), 267276.

  • [8]

    Huppert, B., Endliche Gruppen I, Springer Verlag (Berlin/ Heidelberg/New York, 1979).

  • [9]

    Li, C. and Qiao, S., On Weakly -subgroups and p-nilpotency of finite groups, J. Algebra. Appl., 16, no. 3 (2017) 1750042.

  • [10]

    Shemetkov, L. A., Formations of Finite Groups, (Nauka), (Moscow, 1978).

  • [11]

    Su, N. and Wang, Y., On c-normal and hypercentrally embedded subgroups of finite groups, Algebra Discrete Math. 19 (2015), no. 2, 270282.

    • Search Google Scholar
    • Export Citation
  • [12]

    Wang, Y., On c-normality and its properties, J. Algebra, 180 (1996), 954965.

  • [13]

    Weinstein, M., Between Nilpotent and Solvable, Polygonal House (Passaic, 1982).