Algebra and Logic

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

Finite Generalized Soluble Groups

  • J. HuangEmail author
  • B. Hu
  • A. N. Skiba

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.


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


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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

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