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On weakly Π-embedded subgroups of finite groups

  • 1.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

  • 2.

    School of Mathematics and Informational Science, Shandong Institute of Business and Technology, Yantai 264005, China

Funds:  Supported by NNSF of China (11371335).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2016.12.001
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  • Corresponding author: ZHANG Li (corresponding author), female, born in 1991, PhD. Research field: Group theory. E-mail: zhang12@mail.ustc.edu.cn
  • Received Date: 08 January 2016
  • Accepted Date: 10 May 2016
  • Rev Recd Date: 10 May 2016
  • Publish Date: 30 December 2016
    Abstract

  • Abstract

    Let G be a finite group and H a subgroup of G. H is called weakly Π-embedded in G if there exists a subgroup pair (T, S), where T is a quasinormal subgroup of G containing HG and S/HG≤H/HG satisfies Π-property in G/HG, such that |G:HT| is a power of a prime and (H∩T)/HG≤S/HG. Here weakly Π-embedded subgroups were used to explore the structure of finite groups. In particular, new criterions of hypercyclically embedded subgroups were obtained.

    Abstract

    Let G be a finite group and H a subgroup of G. H is called weakly Π-embedded in G if there exists a subgroup pair (T, S), where T is a quasinormal subgroup of G containing HG and S/HG≤H/HG satisfies Π-property in G/HG, such that |G:HT| is a power of a prime and (H∩T)/HG≤S/HG. Here weakly Π-embedded subgroups were used to explore the structure of finite groups. In particular, new criterions of hypercyclically embedded subgroups were obtained.
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    • References(22)
  • [1]
    DOERK K, HAWKES T. Finite Soluble Groups[M]. Berlin: Walter de Gruyter, 1992.
    [2]
    GUO W. Structure Theory for Canonical Classes of Finite Groups[M]. Berlin/ Heidelberg: Spring, 2015.
    [3]
    HUPPERT B. Endliche Gruppen Ⅰ[M]. Berlin/ New York: Springer-Verlag, 1967.
    [4]
    LI B. On Π-property and Π-normality of subgroups of finite groups[J]. J Algebra, 2011, 334: 321-337.
    [5]
    CHEN X, GUO W. On Π-supplemented subgroups of a finite group[J]. Comm Algebra, 2016, 44: 731-745.
    [6]
    LI B. Finite groups with Π-supplemented minimal subgroups[J]. Comm Algebra, 2013, 41: 2 060-2 070.
    [7]
    SU N, LI Y, WANG Y. A criterion of p-hypercyclically embedded subgroups of finite groups[J]. J Algebra, 2014, 400: 82-93.
    [8]
    DESKINS W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82: 125-132.
    [9]
    HUPPERT B, BLACKBURN N. Finite Groups Ⅲ[M]. Berlin/ Heidelberg: Springer-Verlag, 1982.
    [10]
    MAIER R, SCHMID P. The embedding of quasinormal subgroups in finite groups[J]. Math Z, 1973, 131: 269-272.
    [11]
    BALLESTER-BOLINCHES A, ESTEBAN-ROMERO R, ASAAD M. Products of Finite Groups[M]. Berlin/ New York: Walter de Gruyter, 2010.
    [12]
    GUO W, SKIBA A N. On FΦ*-hypercentral subgroups of finite groups[J]. J Algebra, 2012, 372: 275-292.
    [13]
    LI B, GUO W. On some open problems related to X-permutability of subgroups[J]. Comm Algebra, 2011, 39: 757-771.
    [14]
    GAGEN T M. Topics in Finite Groups[M]. Melbourne/ New York/ London: Cambridge, 1976.
    [15]
    GUO W, SKIBA A N. On the intersection of the F-maximal subgroups and the generalized F-hypercentre of a finite group[J]. J Algebra, 2012, 366: 112-125.
    [16]
    GUO W. The Theory of Classes of Groups[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Science Press/ Kluwer Academic Publishers, 2000.
    [17]
    WANG Y. c-Normality of groups and its properties[J]. J Algebra, 1996, 180: 954-965.
    [18]
    CHEN X, MAO Y, GUO W. On finite groups with some primary subgroups satisfying partial S-Π-property[J]. Comm Algebra, 2017, 45: 428-436.
    [19]
    EZQUERRO L M. A contribution to the theory of finite supersolvable group[J]. Rend Sem Mat Univ Padova, 1993, 89: 161-170.
    [20]
    LI Y, WANG Y, WEI H. The influence of π-quasinormality of some subgroups of a finite group[J]. Arch Math (Basel), 2003, 81: 245-252.
    [21]
    SRINIVASAN S. Two sufficient conditions for supersolvability of finite groups[J]. Israel J Math, 1980, 35: 210-214.
    [22]
    WEI H, WANG Y, LI Y. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II[J]. Comm Algebra, 2003, 31: 4 807-4 816.
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    [1]
    DOERK K, HAWKES T. Finite Soluble Groups[M]. Berlin: Walter de Gruyter, 1992.
    [2]
    GUO W. Structure Theory for Canonical Classes of Finite Groups[M]. Berlin/ Heidelberg: Spring, 2015.
    [3]
    HUPPERT B. Endliche Gruppen Ⅰ[M]. Berlin/ New York: Springer-Verlag, 1967.
    [4]
    LI B. On Π-property and Π-normality of subgroups of finite groups[J]. J Algebra, 2011, 334: 321-337.
    [5]
    CHEN X, GUO W. On Π-supplemented subgroups of a finite group[J]. Comm Algebra, 2016, 44: 731-745.
    [6]
    LI B. Finite groups with Π-supplemented minimal subgroups[J]. Comm Algebra, 2013, 41: 2 060-2 070.
    [7]
    SU N, LI Y, WANG Y. A criterion of p-hypercyclically embedded subgroups of finite groups[J]. J Algebra, 2014, 400: 82-93.
    [8]
    DESKINS W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82: 125-132.
    [9]
    HUPPERT B, BLACKBURN N. Finite Groups Ⅲ[M]. Berlin/ Heidelberg: Springer-Verlag, 1982.
    [10]
    MAIER R, SCHMID P. The embedding of quasinormal subgroups in finite groups[J]. Math Z, 1973, 131: 269-272.
    [11]
    BALLESTER-BOLINCHES A, ESTEBAN-ROMERO R, ASAAD M. Products of Finite Groups[M]. Berlin/ New York: Walter de Gruyter, 2010.
    [12]
    GUO W, SKIBA A N. On FΦ*-hypercentral subgroups of finite groups[J]. J Algebra, 2012, 372: 275-292.
    [13]
    LI B, GUO W. On some open problems related to X-permutability of subgroups[J]. Comm Algebra, 2011, 39: 757-771.
    [14]
    GAGEN T M. Topics in Finite Groups[M]. Melbourne/ New York/ London: Cambridge, 1976.
    [15]
    GUO W, SKIBA A N. On the intersection of the F-maximal subgroups and the generalized F-hypercentre of a finite group[J]. J Algebra, 2012, 366: 112-125.
    [16]
    GUO W. The Theory of Classes of Groups[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Science Press/ Kluwer Academic Publishers, 2000.
    [17]
    WANG Y. c-Normality of groups and its properties[J]. J Algebra, 1996, 180: 954-965.
    [18]
    CHEN X, MAO Y, GUO W. On finite groups with some primary subgroups satisfying partial S-Π-property[J]. Comm Algebra, 2017, 45: 428-436.
    [19]
    EZQUERRO L M. A contribution to the theory of finite supersolvable group[J]. Rend Sem Mat Univ Padova, 1993, 89: 161-170.
    [20]
    LI Y, WANG Y, WEI H. The influence of π-quasinormality of some subgroups of a finite group[J]. Arch Math (Basel), 2003, 81: 245-252.
    [21]
    SRINIVASAN S. Two sufficient conditions for supersolvability of finite groups[J]. Israel J Math, 1980, 35: 210-214.
    [22]
    WEI H, WANG Y, LI Y. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II[J]. Comm Algebra, 2003, 31: 4 807-4 816.
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