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Solubility of finite groups

  • 1.

    Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

  • 2.

    College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, China

Funds:  Supported by a NNSF of China (11371335,11471055).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.06.003
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  • Corresponding author: ZHANG Li (corresponding author), female, born in 1991, PhD. Research field: group theory.
  • Received Date: 21 January 2015
  • Accepted Date: 13 April 2015
  • Rev Recd Date: 13 April 2015
  • Publish Date: 30 June 2015
    Abstract

  • Abstract

    Let H be a p-subgroup of G. Then: ① H satisfies Φ*-property in G if H is a Sylow subgroup of some subnormal subgroup of G and for any non-solubly-Frattini chief factor L/K of G, |G:NG(K(H∩L))| is a power of p; ② H is called Φ*-embedded in G if there exists a subnormal subgroup T of G such that HT is S-quasinormal in G and H∩T≤S, where S≤H satisfies Φ*-property in G. Here Φ*-embedded subgroups were used to study the structure of finite groups and, in particular, some new characterizations for a group G to be soluble are obtained.

    Abstract

    Let H be a p-subgroup of G. Then: ① H satisfies Φ*-property in G if H is a Sylow subgroup of some subnormal subgroup of G and for any non-solubly-Frattini chief factor L/K of G, |G:NG(K(H∩L))| is a power of p; ② H is called Φ*-embedded in G if there exists a subnormal subgroup T of G such that HT is S-quasinormal in G and H∩T≤S, where S≤H satisfies Φ*-property in G. Here Φ*-embedded subgroups were used to study the structure of finite groups and, in particular, some new characterizations for a group G to be soluble are obtained.
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    • References(15)
  • [1]
    Doerk K, Hawkes T. Finite Soluble Groups[M]. Berlin: Walter de Gruyter, 1992.
    [2]
    Guo W. The Theory of Classes of Groups[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Science Press-Kluwer Academic Publishers, 2000.
    [3]
    Huppert B. Endliche Gruppen Ⅰ[M]. Berlin/ Heidelberg/ New York: Springer-Verlag, 1967.
    [4]
    Hall P. A characteristic property of soluble groups[J]. J London Math Soc, 1937, 12:198-200.
    [5]
    Srinivasan S. Two sufficient conditions for supersolubility of finite groups[J]. Israel J Math, 1980, 3: 210-214.
    [6]
    Guo W, Skiba A N. Finite groups with given s-embedded and n-embedded subgroups[J]. J Algebra, 2009, 321: 2 843-2 860.
    [7]
    Guo W, Shum K P, Skiba A N. On solubility and supersolubility of some classes of finite groups[J]. Sci China Ser A, 2009, 52: 272-286.
    [8]
    Guo W, Skiba A N. On FΦ*-hypercentral subgroups of finite groups[J]. J Algebra, 2012, 372: 275-292.
    [9]
    Deskins W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82: 125-132.
    [10]
    Schmid P. Sugroup permutable with all Sylow subgroups[J]. J Algebra, 1998, 207: 285-293.
    [11]
    Wang Y. c-normality of groups and its properties[J]. J Algebra, 1996, 180: 954-965.
    [12]
    Skiba A N. On weakly s-permutable subgroups of finite groups[J]. J Algebra, 2007, 315: 192-209.
    [13]
    Wielandt H. Subnormal subgroups and permutation groups[C]// Lectures given at the Ohio State University. Columbus, Ohio: Ohio State University, 1971.
    [14]
    Chen X, Guo W. On weakly s-embedded and weakly τ-embedded subgroups[J]. Siberian Math J, 2013, 54(5): 931-945.
    [15]
    Robinson D J S. A Course in the Theory of Groups[M]. New York: Springer, 1982.)
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    [1]
    Doerk K, Hawkes T. Finite Soluble Groups[M]. Berlin: Walter de Gruyter, 1992.
    [2]
    Guo W. The Theory of Classes of Groups[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Science Press-Kluwer Academic Publishers, 2000.
    [3]
    Huppert B. Endliche Gruppen Ⅰ[M]. Berlin/ Heidelberg/ New York: Springer-Verlag, 1967.
    [4]
    Hall P. A characteristic property of soluble groups[J]. J London Math Soc, 1937, 12:198-200.
    [5]
    Srinivasan S. Two sufficient conditions for supersolubility of finite groups[J]. Israel J Math, 1980, 3: 210-214.
    [6]
    Guo W, Skiba A N. Finite groups with given s-embedded and n-embedded subgroups[J]. J Algebra, 2009, 321: 2 843-2 860.
    [7]
    Guo W, Shum K P, Skiba A N. On solubility and supersolubility of some classes of finite groups[J]. Sci China Ser A, 2009, 52: 272-286.
    [8]
    Guo W, Skiba A N. On FΦ*-hypercentral subgroups of finite groups[J]. J Algebra, 2012, 372: 275-292.
    [9]
    Deskins W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82: 125-132.
    [10]
    Schmid P. Sugroup permutable with all Sylow subgroups[J]. J Algebra, 1998, 207: 285-293.
    [11]
    Wang Y. c-normality of groups and its properties[J]. J Algebra, 1996, 180: 954-965.
    [12]
    Skiba A N. On weakly s-permutable subgroups of finite groups[J]. J Algebra, 2007, 315: 192-209.
    [13]
    Wielandt H. Subnormal subgroups and permutation groups[C]// Lectures given at the Ohio State University. Columbus, Ohio: Ohio State University, 1971.
    [14]
    Chen X, Guo W. On weakly s-embedded and weakly τ-embedded subgroups[J]. Siberian Math J, 2013, 54(5): 931-945.
    [15]
    Robinson D J S. A Course in the Theory of Groups[M]. New York: Springer, 1982.)
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