Finite groups whose n-maximal subgroups are σ-subnormal

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（7）
• 页码范围: 1355-1372
• 页数: 18页
• 语种: 英文

摘要

Let σ ={σi | 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 Hall σ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G,for some i ∈ I,and H contains exactly one Hall σi-subgroup of G for every σi ∈ 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 HAx =AxH for all A ∈ H and x ∈ G:σ-subnormal in G if there is a subgroup chain A =Ao ≤ A1 ≤...≤ At =G such that either Ai-1 (⊿) Ai or Ai/(Ai-1)Ai is a finite σi-group for some σi ∈ σ for all i =1,...,t.If Mn ＜ Mn-1 ＜ … ＜ M1 ＜ M0 =G,where Mi is a maximal subgroup of Mi-1,i =1,2,...,n,then Mn 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 (mσq(G) =n,respectively).In this paper,we show that the parameters mσ(G) and mσq(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 |π(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 mσ(G) =|π(G)|.Some known results are generalized.