2018/01/28 by Bin Hu, Hu, Bin, Jianhong Huang +3
Mathematics · Engineering · #Finite Group Theory Research #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1801.09234
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=\bigcupi∈ 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\trianglelefteq 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)\rtimes (G/CG(H/K)) is σ-primary.