In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems ab...In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems about nilpotent subgroup.展开更多
According to Hall,a subgroup H of a group G is said to be pronormal if H and H^(g) are conjugate in<H,H^(g)>for every g∈G.In this survey,we discuss the role of pronormality for some subgroups of finite groups:H...According to Hall,a subgroup H of a group G is said to be pronormal if H and H^(g) are conjugate in<H,H^(g)>for every g∈G.In this survey,we discuss the role of pronormality for some subgroups of finite groups:Hall subgroups,subgroups of odd index,submaximal X-subgroup,etc.展开更多
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
文摘In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems about nilpotent subgroup.
基金supported by a NNSF Grant of China(Grant#11771409)Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciencesby SB RAS Fundamental Research Program I.1.1(project#0314-2016-0001).
文摘According to Hall,a subgroup H of a group G is said to be pronormal if H and H^(g) are conjugate in<H,H^(g)>for every g∈G.In this survey,we discuss the role of pronormality for some subgroups of finite groups:Hall subgroups,subgroups of odd index,submaximal X-subgroup,etc.
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.
