Advertisement

Algebra and Logic

, Volume 58, Issue 2, pp 173–185 | Cite as

Finite Generalized Soluble Groups

  • J. HuangEmail author
  • B. Hu
  • A. N. Skiba
Article
  • 18 Downloads

Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. Suppose σ(G) = {σi | σi ∩ π(G) ≠ = ∅}. A set ℋ of subgroups of G is called a complete Hall σ-set of G if every nontrivial member of ℋ is a σi-subgroup of G for some iI and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ∈ σ(G). A group G is σ-full if G possesses a complete Hall σ-set. A complete Hall σ-set ℋ of G is called a σ-basis of G if every two subgroups A, B ∈ ℋ are permutable, i.e., AB = BA. In this paper, we study properties of finite groups having a σ-basis. It is proved that if G has a σ-basis, then G is generalized σ-soluble, i.e, |σ(H/K)| ≤ 2 for every chief factor H/K of G. Moreover, it is shown that every complete Hall σ-set of a σ-full group G forms a σ-basis of G iff G is generalized σ-soluble, and for the automorphism group G/CG(H/K) induced by G on any its chief factor H/K, we have |σ(G/CG(H/K))| ≤ 2 and also σ(H/K) ⊆ σ(G/CG(H/K)) in the case |σ(G/CG(H/K))| = 2.

Keywords

finite group Hall subgroup σ-soluble subgroup σ-basis generalized σ-soluble group 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. N. Skiba, “On σ-subnormal and σ-permutable subgroups of finite groups,” J. Alg., 436, 1-16 (2015).MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. N. Skiba, “On some results in the theory of finite partially soluble groups,” Commun. Math. Stat., 4, No. 3, 281-309 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. N. Skiba, “A generalization of a Hall theorem,” J. Alg. Appl., 15, No. 5 (2016), Article ID 1650085.Google Scholar
  4. 4.
    A. N. Skiba, “Some characterizations of finite σ-soluble PσT-groups,” J. Alg., 495, 114-129 (2018).MathSciNetCrossRefGoogle Scholar
  5. 5.
    I. M. Isaacs, “Semipermutable π-subgroups,” Arch. Math., 102, No. 1, 1-6 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. T. Borovikov, “Groups with permutable subgroups of mutually simple orders,” Voprosy Alg., Iss. 5, 80–82 (1990).Google Scholar
  7. 7.
    B. Huppert, Endliche Gruppen. I, Grundl. Math. Wissensch. Einzeldarst., 134, Springer, Berlin (1967).Google Scholar
  8. 8.
    X. Yin and N. Yang, “Finite groups with permutable Hall subgroups,” Front. Math. China, 12, No. 5, 1265-1275 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. Zhang, “Sylow numbers of finite groups,” J. Alg., 176, No. 1, 111-123 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    W. Guo, “Finite groups with given indices of normalizers of Sylow subgroups,” Sib. Math. J., 37, 207-214 (1996).Google Scholar
  11. 11.
    K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Expo. Math., 4, W. de Gruyter, Berlin (1992).Google Scholar
  12. 12.
    L. A. Shemetkov and A. N. Skiba, Formations of Algebraic Systems [in Russian], Nauka, Moscow (1989).zbMATHGoogle Scholar
  13. 13.
    A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).Google Scholar
  14. 14.
    A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups,Math. Appl. (Springer), 584, Springer-Verlag, Dordrecht (2006).Google Scholar
  15. 15.
    A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad, Products of Finite Groups, De Gruyter Exp. Math., 53, Walter de Gruyter, Berlin (2010).Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouP.R. China
  2. 2.Department of Mathematics and Technologies of ProgrammingFrancisk Skorina Gomel State UniversityGomelBelarus

Personalised recommendations