Paper «On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups» published in Mathematics

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Abd El-Rahman Heliel, Mohammed Al-Shomrani, Adolfo Ballester-Bolinches.
On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups.
Mathematics, 8(12):2165 (4 pages), 2020.

doi:10.3390/math8122165

Abstract

Let σ={σi:iI} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by (G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by (G). If F is a subgroup-closed saturated formation, we define the σ-F-length (G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then (A,F)=(G,F)−i for some i∈{0,1,2}.

Keywords: finite group; σ-solubility; σ-nilpotency; σ-nilpotent length

Paper «On σ-subnormal closure» published in Comm. Algebra

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M. M. Al-Shomrani, A. A. Heliel, and Adolfo Ballester-Bolinches.
On σ-subnormal closure.
Comm. Algebra, 48(8):3624–3627, 2020.

doi:10.1080/00927872.2020.1742348

Abstract

Let σ = {σi : iI} be a partition of the set ℙ of all prime numbers. A subgroup A of a finite group G is called σsubnormal in G if there is chain of subgroups A = A0A1 ⊆⋯⊆ An = G with Aj-1 normal in Aj or Ai/CoreAi(Ai-1) is a σj-group for some jI, 1 ≤ in. In this paper, the description of the unique smallest σ-subnormal subgroup of a σ-soluble group containing a given subgroup is obtained.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; σ-soluble group; σ-subnormal subgroup