Let a = {σi｜ i ∈ I} be some partition of the set of all primes P, G a finite group and σ（G） = {σi｜σi ∩ π （G） ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠...Let a = {σi｜ i ∈ I} be some partition of the set of all primes P, G a finite group and σ（G） = {σi｜σi ∩ π （G） ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ 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-semipermutablc in G with respect to H if HHi x = Hi x H for all x ∈ G and all x ∈ G and all Hi ∈H such that （｜H｜, ｜Hi｜） = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.展开更多
Throughout this paper, all groups are finite and G always denotes a finite group. We use π（G） to denote the set of all primes dividing ｜G｜. A subgroup A of G is said to permute with a subgroup B if AB = BA. In th...Throughout this paper, all groups are finite and G always denotes a finite group. We use π（G） to denote the set of all primes dividing ｜G｜. A subgroup A of G is said to permute with a subgroup B if AB = BA. In this case, we say also that the subgroups A and B are permutable.展开更多
基金Supported by NNSF(Grant No.11771409)Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences
文摘Let a = {σi｜ i ∈ I} be some partition of the set of all primes P, G a finite group and σ（G） = {σi｜σi ∩ π （G） ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ 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-semipermutablc in G with respect to H if HHi x = Hi x H for all x ∈ G and all x ∈ G and all Hi ∈H such that （｜H｜, ｜Hi｜） = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.
文摘Throughout this paper, all groups are finite and G always denotes a finite group. We use π（G） to denote the set of all primes dividing ｜G｜. A subgroup A of G is said to permute with a subgroup B if AB = BA. In this case, we say also that the subgroups A and B are permutable.
