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
In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble gro...In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.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In this paper, we prove the following theorem： Let p be a prime number, P a Sylow psubgroup of a group G and π = π（G） ／ {p}. If P is seminormal in G, then the following statements hold： 1） G is a p-soluble group and P＇ ≤ Op（G）; 2） lp（G） ≤ 2 and lπ（G） ≤ 2; 3） if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.
