On σ-quasinormal subgroups of finite groups

Abstract

Let G be a finite group and σ = {σi | i ∈ I} some partition of the set of all primes P, that is, P = Si∈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 E Ai or Ai/(Ai−1)Ai is σ-primary for all i = 1, . . . , n; modular in G if the following conditions hold: (i) hX, A ∩ Zi = hX, Ai ∩ Z for all X ≤ G, Z ≤ G such that X ≤ Z and (ii) hA, Y ∩ Zi = hA, Yi ∩ Z for all Y ≤ G, Z ≤ G such that A ≤ Z; and σ-quasinormal in G if A 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 every chief factor H/K of G between HG and HG is σ-central in G, that is, the semidirect product (H/K) o (G/CG(H/K)) is σ-primary.