On σ-quasinormal subgroups of finite groups

Original · EN

Let G be a finite group and σ={σᵢ | i∈ I} some partition of the set of all primes P, that is, σ={σᵢ | i∈ I }, where P=ᵢ∈ ᵢ σᵢ and σᵢ∩ σⱼ= for all i≠ j. We say that G is σ-primary if G is a σᵢ-group for some i. A subgroup A of G is said to be: σ-subnormal in G if there is a subgroup chain A=A₀ ≤ A₁ ≤ ≤ Aₙ=G such that either Aᵢ₋₁ Aᵢ or Aᵢ/(Aᵢ₋₁)ₐᵢ is σ-primary for all i=1,, n, modular in G if the following conditions hold: (i) X, A ∩ Z = X, A ∩ Z for all X ≤ G, Z ≤ G such that X ≤ Z, and (ii) A, Y ∩ Z = A, Y ∩ Z for all Y ≤ G, Z ≤ G such that A ≤ Z. In this paper, a subgroup A of G is called σ-quasinormal in G if L is modular and σ-subnormal in G. We study σ-quasinormal subgroups of G. In particular, we prove that if a subgroup H of G is σ-quasinormal in G, then for every chief factor H/K of G between Hᵍ and HG the semidirect product (H/K) (G/CG(H/K)) is σ-primary.