Finite groups all of whose subgroups are σ-subnormal or σ-abnormal

摘要

Let σ = {σ_i|i ∈ I} be a partition of the set of all primes P and G a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G for some i ∈ I, and H contains exact one Hall σ_i-subgroup of G for every i such that σ_i ∩ π(G) 6= ?. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that AH~x = H~xA for all H ∈ H and all x ∈ G; σ- subnormal in G if there is a subgroup chain A = A_0 ≤ A_1 ≤ · · · ≤ A_t = G such that either A_(i?1) ? A_i or A_i/(A_(i?1))A_i is σ-primary for all i = 1, . . . , t; σ-abnormal in G if L/K_L is not σ-primary whenever A ≤ K < L ≤ G. In this paper, answering to Question 7.7 in [17], we describe finite groups in which every subgroup is either σ-subnormal or σ-abnormal, and we use this result to classify finite groups G such that every subgroup of G is either σ-quasinormal or σ-abnormal in G.