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On the Norm of the Abelian p -Group-Residuals

Author

Listed:
  • Baojun Li

    (School of Sciences, Nantong University, Nantong 226019, China)

  • Yu Han

    (School of Sciences, Nantong University, Nantong 226019, China)

  • Lü Gong

    (School of Sciences, Nantong University, Nantong 226019, China)

  • Tong Jiang

    (School of Sciences, Nantong University, Nantong 226019, China)

Abstract

Let G be a group. D p ( G ) = ⋂ H ≤ G N G ( H ′ ( p ) ) is defined and, the properties of D p ( G ) are investigated. It is proved that D p ( G ) = P [ A ] , where P = D ( P ) is the Sylow p -subgroup and A = N ( A ) is a Hall p ′ -subgroup of D p ( G ) , respectively. Furthermore, it is proved in a group G that (1) D p ( G ) = 1 if and only if C G ( G ′ ( p ) ) = 1 ; (2) O p ′ ( D p ( G ) ) ≤ Z ∞ ( O p ( G ) ) and (3) if Z ( G ′ ( p ) ) = 1 , then C G ( G ′ ( p ) ) = D p ( G ) .

Suggested Citation

  • Baojun Li & Yu Han & Lü Gong & Tong Jiang, 2021. "On the Norm of the Abelian p -Group-Residuals," Mathematics, MDPI, vol. 9(8), pages 1-6, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:8:p:842-:d:534909
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