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The product of two σ-supersoluble groups

  • MAO Yuemei, MA Xiaojian

Funds:  Supported by Youth Program of National Natural Science Foundation of China (11901364), Applied Basic Research Program Project of Shanxi Province (201901D211439), Applied Basic Research Project of Datong City (2019156), Science and Technology Innovation Project of Shanxi Province (2019L0747).
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https://doi.org/10.3969/j.issn.0253-2778.2020.04.004
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  • Corresponding author: MAO Yuemei(corresponding author)，female, born in 1979, PhD/ associated Prof. Research field: Theory of groups.E-mail:maoyuemei@126.com
  • Received Date: 12 March 2020
  • Accepted Date: 15 April 2020
  • Rev Recd Date: 15 April 2020
  • Publish Date: 30 April 2020
    Abstract

  • Abstract

    Let Nσ denote the classes of all σ-nilpotent groups and GNσ be the σ-nilpotent residual of G. We say that G is σ-supersoluble if each chief factor of G below GNσ is cyclic. A subgroup H of G is said to be completely c-permutable with a subgroup T of G if there exists an element x∈〈H,T〉 such that HTx=TxH.

    Abstract

    Let Nσ denote the classes of all σ-nilpotent groups and GNσ be the σ-nilpotent residual of G. We say that G is σ-supersoluble if each chief factor of G below GNσ is cyclic. A subgroup H of G is said to be completely c-permutable with a subgroup T of G if there exists an element x∈〈H,T〉 such that HTx=TxH.
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    • References(12)
  • [1]
    SKIBA A N. On σ-subnormal and σ-permutable subgroups of finite groups[J]. Journal of Algebra, 2015, 436:1-16.
    [2]
    SKIBA A N. On some results in the theory of finite partially soluble groups[J]. Communications in Mathematics and Statistics, 2016, 4(3): 281-309.
    [3]
    GUO W, ZHANG C, SKIBA A N. On σ-supersoluble groups and one generalization of CLT-groups[J]. Journal of Algebra, 2018,512: 92-108.
    [4]
    GUO W, SHUM K P, SKIBA A N. Conditionally permutable subgroups and supersolubility of finite groups[J]. Southeast Asian Bulletin of Mathematics, 2018, 29(3): 493-510.
    [5]
    LIU Xi, GUO W, SHUM K P. Products of finite supersoluble groups[J]. Algebra Colloquium, 2009, 16(2): 333-340.
    [6]
    DOERK K, HAWKES T. Finite Soluble Groups[M].Berlin: Walter de Gruyter, 1992.
    [7]
    GUO W. Structure Theory for Canonical Class of Finite Groups[M]. Heidelberg: Springer, 2015.
    [8]
    HUPPERT B. Endliche Gruppen Ⅰ[M]. New York: Springer, 1967.
    [9]
    SKIBA A N. A generalization of a Hall theorem[J]. Journal of Algebra and Its Applications, 2016, 15(5): 207-214.
    [10]
    KNYAGINA V N, MONAKHOV V S. On π′-properties of finite groups having a Hall π-subgroup[J]. Siberian Mathematical Journal, 2011, 52(2): 297-309.
    [11]
    GUO W, SKIBA A N. On Π-quasinormal subgroups of finite groups[J]. Monatshefte für Mathematik, 2018, 185(3): 443-453.
    [12]
    BALLESTER-BOLINCHES A, ESTEBAN-ROMERO R, ASAAD M. Products of Finite Groups[M]. Berlin: Walter de Gruyter, 2010.)
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    [1]
    SKIBA A N. On σ-subnormal and σ-permutable subgroups of finite groups[J]. Journal of Algebra, 2015, 436:1-16.
    [2]
    SKIBA A N. On some results in the theory of finite partially soluble groups[J]. Communications in Mathematics and Statistics, 2016, 4(3): 281-309.
    [3]
    GUO W, ZHANG C, SKIBA A N. On σ-supersoluble groups and one generalization of CLT-groups[J]. Journal of Algebra, 2018,512: 92-108.
    [4]
    GUO W, SHUM K P, SKIBA A N. Conditionally permutable subgroups and supersolubility of finite groups[J]. Southeast Asian Bulletin of Mathematics, 2018, 29(3): 493-510.
    [5]
    LIU Xi, GUO W, SHUM K P. Products of finite supersoluble groups[J]. Algebra Colloquium, 2009, 16(2): 333-340.
    [6]
    DOERK K, HAWKES T. Finite Soluble Groups[M].Berlin: Walter de Gruyter, 1992.
    [7]
    GUO W. Structure Theory for Canonical Class of Finite Groups[M]. Heidelberg: Springer, 2015.
    [8]
    HUPPERT B. Endliche Gruppen Ⅰ[M]. New York: Springer, 1967.
    [9]
    SKIBA A N. A generalization of a Hall theorem[J]. Journal of Algebra and Its Applications, 2016, 15(5): 207-214.
    [10]
    KNYAGINA V N, MONAKHOV V S. On π′-properties of finite groups having a Hall π-subgroup[J]. Siberian Mathematical Journal, 2011, 52(2): 297-309.
    [11]
    GUO W, SKIBA A N. On Π-quasinormal subgroups of finite groups[J]. Monatshefte für Mathematik, 2018, 185(3): 443-453.
    [12]
    BALLESTER-BOLINCHES A, ESTEBAN-ROMERO R, ASAAD M. Products of Finite Groups[M]. Berlin: Walter de Gruyter, 2010.)
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