Advertisement

On the Solvability of a Finite Group with S-Seminormal Schmidt Subgroups

  • V. N. Knyagina
  • V. S. Monakhov
  • E. V. Zubei
Article
  • 1 Downloads

A finite nonnilpotent group is called a Schmidt group if all its proper subgroups are nilpotent. A subgroup A is called S-seminormal (or SS-permutable) in a finite group G if there is a subgroup B such that G = AB and A is permutable with every Sylow subgroup of B. We establish criteria for the solvability and 𝜋-solvability of finite groups in which some Schmidt subgroups are S-seminormal. In particular, we prove the solvability of a finite group in which all supersoluble Schmidt subgroups of even order are S-seminormal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Yu. Schmidt, “Groups all subgroups of which are special,” Mat. Sb., 31, 366–372 (1924).Google Scholar
  2. 2.
    N. F. Kuzennyi and S. S. Levishchenko, “Finite Schmidt’s groups and their generalizations,” Ukr. Mat. Zh., 43, No. 7-8, 963–968 (1991); English translation: Ukr. Math. J., 43, No. 7-8, 904–908 (1991).Google Scholar
  3. 3.
    V. S. Monakhov, “Schmidt subgroups, their existence, and some applications,” in: Proceedings of the Ukrainian Mathematical Congress, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, No. 1 (2002), pp. 81–90.Google Scholar
  4. 4.
    V. N. Knyagina and V. S. Monakhov, “On finite groups with some subnormal Schmidt subgroups,” Sib. Mat. Zh., 45, No. 6, 1316–1322 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. A.Vedernikov, “Finite groups with subnormal Schmidt subgroups,” Algebra Logika, 46, No. 6, 669–687 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kh. A. Al-Sharo and A. N. Skiba, “On finite groups with σ-subnormal Schmidt subgroups,” Comm. Algebra, 45, 4158–4165 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. N. Knyagina and V. S. Monakhov, “Finite groups with Hall–Schmidt subgroups,” Publ. Math. Debrecen., 81, No. 3-4, 341–350 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    X. Su, “On semi-normal subgroups of finite group,” J. Math. (Wuhan), 8, No. 1, 7–9 (1988).Google Scholar
  9. 9.
    A. Carocca and H. Matos, “Some solvability criteria for finite groups,” Hokkaido Math. J., 26, No. 1, 157–161 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. V. Podgornaya, “Seminormal subgroups and supersolvability of finite groups,” Vests. Nats. Akad. Nauk Belarus., Ser. Fiz.-Mat. Navuk., No. 4, 22–25 (2000).MathSciNetGoogle Scholar
  11. 11.
    V. S. Monakhov, “Finite groups with seminormal Hall subgroup,” Mat. Zametki, 80, No. 4, 573–581 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. N. Knyagina and V. S. Monakhov, “Finite groups with seminormal Schmidt subgroups,” Algebra Logika, 46, No. 4, 448–458 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad, Products of Finite Groups, de Gruyter, Berlin (2010).CrossRefzbMATHGoogle Scholar
  14. 14.
    O. Kegel, “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z., 78, 205–221 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Assad and A. A. Heliel, “On S-quasinormally embedded subgroups of finite groups,” J. Pure Appl. Algebra, 165, 129–135 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    W. Guo, K. P. Shum, and A. N. Skiba, “X-permutable maximal subgroups of Sylow subgroups of finite groups,” Ukr. Mat. Zh., 58, No. 10, 1299–1309 (2006); English translation: Ukr. Math. J., 58, No. 10, 1471–1480 (2006).Google Scholar
  17. 17.
    W. Guo, K. P. Shum, and A. N. Skiba, “X-semipermutable subgroups of finite groups,” J. Algebra, 315, 31–41 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    S. R. Li, Z. C. Shen, and X. H. Kong, “On SS-quasinormal subgroups of finite groups,” Comm. Algebra, 36, No. 12, 4436–4447 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. R. Li, Z. C. Shen, J. J. Liu, and X. C. Liu, “The influence of SS-quasinormality of some subgroups on the structure of finite groups,” J. Algebra, 319, 4275–4287 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    I. M. Isaacs, “Semipermutable 𝜋-subgroups,” Arch. Math., 102, 1–6 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    B. Huppert, Endliche Gruppen I, Springer, Berlin (1967).CrossRefzbMATHGoogle Scholar
  22. 22.
    V. S. Monakhov, Introduction to the Theory of Finite Groups and Their Classes [in Russian], Vysheishaya Shkola, Minsk (2006).Google Scholar
  23. 23.
    V. S. Monakhov, “On finite groups with given collection of Schmidt subgroups,” Mat. Zametki, 58, No. 5, 717–722 (1995).MathSciNetGoogle Scholar
  24. 24.
    Ya. G. Berkovich, “Theorem on nonnilpotent solvable subgroups of a finite group,” in: Finite Groups [in Russian], Nauka i Tekhnika, Minsk (1966), pp 24–39.Google Scholar
  25. 25.
    V. S. Monakhov, “On Schmidt subgroups of finite groups,” Vopr. Algebr., Issue 13, 153–171 (1998).Google Scholar
  26. 26.
    V. S. Monakhov, “On the product of a 2-decomposable group and a Schmidt group,” Dokl. Akad. Nauk Belorus. SSR, 18, No. 10, 871–874 (1974).Google Scholar
  27. 27.
    D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, New York (1982).zbMATHGoogle Scholar
  28. 28.
    B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin (1982).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. N. Knyagina
    • 1
  • V. S. Monakhov
    • 1
  • E. V. Zubei
    • 1
  1. 1.Skorina Gomel’ UniversityGomelBelarus

Personalised recommendations