Algebra and Logic

, Volume 58, Issue 2, pp 173–185

# Finite Generalized Soluble Groups

• J. Huang
• B. Hu
• A. N. Skiba
Article

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

## References

1. 1.
A. N. Skiba, “On σ-subnormal and σ-permutable subgroups of finite groups,” J. Alg., 436, 1-16 (2015).
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).
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).
5. 5.
I. M. Isaacs, “Semipermutable π-subgroups,” Arch. Math., 102, No. 1, 1-6 (2014).
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).
9. 9.
J. Zhang, “Sylow numbers of finite groups,” J. Alg., 176, No. 1, 111-123 (1995).
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).
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