Archivo por meses: septiembre 2020

Paper «Products of finite connected subgroups» published in Mathematics

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María Pilar Gállego, Peter Hauck, Lev S. Kazarin, Ana Martínez-Pastor, and María Dolores Pérez-Ramos.
Products of finite connected subgroups.
Mathematics, 18(9):1498 (8 pages), 2020.

doi:10.3390/math8091498

Abstract

For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

2020 Mathematics Subject Classification: 20D40, 20E45, 20D20, 20D60

Keywords: finite groups; products of subgroups; two-generated subgroups; L-connection; Fitting classes; Fitting series; formations

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 : i ∈ I} be a partition of the set ℙ of all prime numbers. A subgroup A of a finite group G is called σ–subnormal in G if there is chain of subgroups A = A0 ⊆ A1 ⊆⋯⊆ An = G with Aj-1 normal in Aj or Ai/CoreAi(Ai-1) is a σj-group for some j ∈ I, 1 ≤ i ≤ n. 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

Paper «On a paper of Beltrán and Shao about coprime action» published in J. Pure Appl. Algebra

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H. Meng and A. Ballester-Bolinches.
On a paper of Beltrán and Shao about coprime action.
J. Pure Appl. Algebra, 224(8):106313, 4, 2020.

doi:10.1016/j.jpaa.2020.106313

Abstract

Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Itô about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.

2020 Mathematics Subject Classification: 20D10, 20D25

Keywords: finite groups; coprime action; solubility; p-nilpotency