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X-quasinormal subgroups

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Abstract

Considering two subgroups A and B of a group G and ⊘ ≠ X ⊆ G, we say that A is X-permutable with B if AB x = B x A for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.

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References

  1. Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin; New York (1992).

    MATH  Google Scholar 

  2. Ore O., “Contributions in the theory of groups of finite order,” Duke Math. J., 5, 431–460 (1939).

    Article  MATH  Google Scholar 

  3. Stonehewer S. E., “Permutable subgroups of infinite groups,” Math. Z., 125, No. 1, 1–16 (1972).

    Article  MATH  Google Scholar 

  4. Ito N. and Szep J., “Uber die Quasinormalteiler von endlichen Gruppen,” Acta Sci. Math., 23, No. 1–2, 168–170 (1962).

    MATH  Google Scholar 

  5. Maier R. and Schmid P., “The embedding of quasinormal subgroups in finite groups,” Math. Z., 131, No. 3, 269–272 (1973).

    Article  MATH  Google Scholar 

  6. Kegel O., “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z., 78, No. 1, 205–221 (1962).

    Article  MATH  Google Scholar 

  7. Deskins W. E., “On quasinormal subgroups of finite groups,” Math. Z., 82, No. 2, 125–132 (1963).

    Article  MATH  Google Scholar 

  8. Schmid P., “Subgroups permutable with all Sylow subgroups,” J. Algebra, 207, No. 1, 285–293 (1998).

    Article  MATH  Google Scholar 

  9. Polyakov L. Ya., “Finite groups with permutable subgroups,” in: Finite Groups [in Russian], Nauka i Tekhnika, Minsk, 1966, pp. 75–88.

    Google Scholar 

  10. Maier R., “Endliche metanilpotente Gruppen,” Arch. Math., 23, No. 1, 139–144 (1972).

    Article  MATH  Google Scholar 

  11. Maier R., “Zur Vertauschbarkeit und Subnormalitat von Untergruppen,” Arch. Math., 53, No. 2, 110–120 (1989).

    Article  MATH  Google Scholar 

  12. Su H., “Semi-normal subgroups of finite groups,” Math. Mag., 8, No. 1, 7–9 (1988).

    Google Scholar 

  13. Wang R., “Some sufficient conditions of a nilpotent group,” J. Algebra, 148, No. 2, 289–294 (1992).

    Article  MATH  Google Scholar 

  14. Foguel T., “On seminormal subgroups,” J. Algebra, 165, No. 3, 633–635 (1994).

    Article  MATH  Google Scholar 

  15. Podgornaya V. V., “Seminormal subgroups and supersolubility of finite groups,” Vestsi Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, 4, No. 4, 22–25 (2000).

  16. Skiba A. N., “H-permutable subgroups,” Izv. F. Skorina Gomel State Univ., No. 4, 37–39 (2003).

  17. Guo W., Shum K. P., and Skiba A. N., X-Permutable Subgroups [Preprint, No. 61], F. Skorina Gomel State Univ., Gomel (2003).

    Google Scholar 

  18. Guo W., Shum K. P., and Skiba A. N., Conditionally Permutable Subgroups [Preprint, No. 10], F. Skorina Gomel State Univ., Gomel (2002).

    Google Scholar 

  19. Guo Wenbin, Shum Kar-Ping, and Skiba A. N., “G-Covering systems of subgroups for classes of p-supersoluble and p-nilpotent finite groups,” Siberian Math. J., 45, No. 3, 433–442 (2004).

    Article  Google Scholar 

  20. Guo W., Shum K. P., and Skiba A. N., “Conditionally permutable subgroups and supersolubility of finite groups,” Southeast Asian Bull Math., 29, No. 2, 240–254 (2005).

    Google Scholar 

  21. Guo W., Shum K. P., and Skiba A. N., “Criterions of supersolubility for products of supersoluble groups,” Publ. Math. Debrecen., 68, No. 3–4, 433–449 (2006).

    MATH  Google Scholar 

  22. Al-Sheikahmad A., “Finite groups with given c-permutable subgroups,” Algebra Discrete Math., 3, No. 3, 9–14 (2004).

    Google Scholar 

  23. Hall P., “On the Sylow systems of a soluble group,” Proc. London Math. Soc., 43,No. 2, 507–528 (1937).

    MATH  Google Scholar 

  24. Arad Z. and Ward Michael B., “New criteria for the solvability of finite groups,” J. Algebra, 77, No. 1, 234–246 (1982).

    Article  MATH  Google Scholar 

  25. Huppert B. and Blackburn N., Finite Groups. III, Springer-Verlag, Berlin; New York (1982).

    MATH  Google Scholar 

  26. Shemetkov L. A., Formations of Finite Groups [in Russian], Nauka, Moscow (1978).

    MATH  Google Scholar 

  27. Huppert B., “Zur Sylowstruktur auflosbarer Gruppen,” Arch. Math., 12, No. 1, 161–169 (1961).

    Article  MATH  Google Scholar 

  28. Skiba A. N., “Finite groups with given systems of generalized permutable subgroups,” Izv. F. Skorina Gomel State Univ., No. 3, 12–31 (2006).

  29. Guo W., Shum K. P., and Skiba A., “G-covering subgroup systems for the classes of supersoluble and nilpotent groups,” Israel J. Math., 138, No. 1, 125–138 (2003).

    MATH  Google Scholar 

  30. Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin; Heidelberg; New York (1967).

    MATH  Google Scholar 

  31. Kegel O., “Producte nilpotenter Gruppen,” Arch. Math., 12, No. 1, 90–93 (1961).

    Article  MATH  Google Scholar 

  32. Monakhov V. S., “The product of a supersoluble cyclic or primary group,” in: Finite Groups [in Russian] (Trudy Gomel Sem., Gomel’, 1975–1977), Nauka i Tekhnika, Minsk, 1978, pp. 50–63.

    Google Scholar 

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Authors and Affiliations

  1. Xuzhou Normal University, Xuzhou, P. R. China

    W. Guo

  2. The Chinese University of Hong Kong, Hong Kong, P. R. China

    K. P. Shum

  3. F. Scorina Gomel State University, Gomel’, Belarus

    A. N. Skiba

Authors
  1. W. Guo
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  2. K. P. Shum
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  3. A. N. Skiba
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__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 4, pp. 742–759, July–August, 2007.

Original Russian Text Copyright © 2007 Guo W., Skiba A. N., and Shum K. P.

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Guo, W., Shum, K.P. & Skiba, A.N. X-quasinormal subgroups. Sib Math J 48, 593–605 (2007). https://doi.org/10.1007/s11202-007-0061-x

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  • DOI: https://doi.org/10.1007/s11202-007-0061-x

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