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 con...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 = A0≤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.展开更多
基金supported by National Nature Science Foundation of China (Grant No. 11771409)Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences
文摘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 = A0≤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.
