Finite groups whose n-maximal subgroups are σ-subnormal

Abstract

Let σ = {σ_i | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hallσ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G, for some i ∈ I, and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HA^x = A^xH for all A ∈ H and x ∈ G: σ-subnormal in G if there is a subgroup chain A = A₀ ≤ A₁ ≤ · · · ≤ A_t = G such that either ${A_{i - 1}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } {A_i}$ or A_i=(A_i-1)Ai is a finite σ_i-group for some σ_i ∈ σ for all i = 1;:::; t.

If M_n < M_n-1 < · · · < M₁ < M₀ = G, where Mi is a maximal subgroup of M_i-1, i = 1; 2;...; n, then M_n is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal (σ-quasinormal, respectively) in G but, in the case n > 1, some (n−1)-maximal subgroup is not σ-subnormal (not σ-quasinormal, respectively) in G, we write mσ(G) = n (m_σq(G) = n, respectively).

In this paper, we show that the parameters m_σ(G) and m_σq(G) make possible to bound the σ-nilpotent length lσ(G) (see below the definitions of the terms employed), the rank r(G) and the number |П(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m_σ(G) = |П(G)|. Some known results are generalized.