Finite groups with given systems of generalised σ-permutable subgroups

DOI: https://doi.org/10.33581/2520-6508-2021-3-25-33

Abstract

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 = < A, B >; (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.

Author Biography

Viktoria S. Zakrevskaya, Francisk Skorina Gomel State University, 104 Savieckaja Street, Homieĺ 246019, Belarus

postgraduate student at the department of algebra and geometry, faculty of mathematics and technologies of programming

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Published
2021-12-16
Keywords: finite group, σ-soluble groups, σ-nilpotent group, partially σ-permutable subgroup, (𝔄, σ)-embedded subgroup, 𝔄-normal subgroup
How to Cite
Zakrevskaya, V. S. (2021). Finite groups with given systems of generalised σ-permutable subgroups. Journal of the Belarusian State University. Mathematics and Informatics, 3, 25-33. https://doi.org/10.33581/2520-6508-2021-3-25-33
Issue
No 3 (2021): Journal of the Belarusian State University. Mathematics and Informatics
Section
Mathematical Logic, Algebra and Number Theory

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