Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing │H│, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal sub...Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing │H│, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G; H is called c^＊-quasinormally embedded in G if there is a subgroup T of G such that G = HT and HCqT is s-quasinormally embedded in G. We investigate the influence of c^＊-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.展开更多
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This work was supported in part by the National Natural Science Foundation of China （Grant No. 11071229） and the Natural Science Foundation the Jiangsu Higher Education Institutions （Grant No. J0KJD110004）.
文摘Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing │H│, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G; H is called c^＊-quasinormally embedded in G if there is a subgroup T of G such that G = HT and HCqT is s-quasinormally embedded in G. We investigate the influence of c^＊-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.
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.
