Finite groups with given systems of generalised σ-permutable subgroups
| Autor: | Viktoria S. Zakrevskaya |
|---|---|
| Jazyk: | Belarusian<br />English<br />Russian |
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Журнал Белорусского государственного университета: Математика, информатика, Iss 3, Pp 25-33 (2021) |
| Druh dokumentu: | article |
| ISSN: | 2520-6508 2617-3956 |
| DOI: | 10.33581/2520-6508-2021-3-25-33 |
| Popis: | Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. 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 i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all 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 σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = ; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised. |
| Databáze: | Directory of Open Access Journals |
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