On Π-quasinormal subgroups of finite groups

• Published in: Monatshefte für Mathematik 2018, Vol.185 (3), p.443-453
• Main Authors: Guo, Wenbin, Skiba, Alexander N.
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics; Subgroups
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-016-1007-9

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P and Π a non-empty subset of the set σ . A set H of subgroups of a finite group G is said to be a complete Hall Π -set of G if every member ≠ 1 of H is a Hall σ i -subgroup of G for some σ i ∈ Π and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ Π such that σ i ∩ π ( G ) ≠ ∅ . A subgroup H of G is called Π - permutable or Π - quasinormal in G if G possesses a complete Hall Π -set H such that A H x = H x A for all H ∈ H and x ∈ G . We study the embedding properties of H under the hypothesis that H is Π -permutable in G . Some well-known results are generalized.