Archivo de la etiqueta: finite group

Paper «On σ-subnormal subgroups of factorised finite groups» published in J. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and X. Yi.
On σ-subnormal subgroups of factorised finite groups.
J. Algebra, 559:195–202, 2020.

doi:10.1016/j.jalgebra.2020.05.002

Abstract

Let σ = {σi : i ∈ I} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σ–subnormal in G if there is chain of subgroups X = X0 ⊆ X1 ⊆⋯⊆ Xn = G with Xj-1 normal in Xj or Xi/CoreXi(Xi-1) is a σ-group for some i ∈ I, 1 ≤ j ≤ n. In the special case that σ is the partition of ℙ into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality.

If a finite soluble group G = AB is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is σ-subnormal in 〈X, Xg〉 for all g ∈ A ∪ B , we prove that X is σ-subnormal in G. This is an extension of a subnormality criteria due to Maier and Sidki and Casolo.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; Soluble group; σ-Subnormal subgroup; σ-Nilpotency; Factorised group

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 products of groups and indices not divisible by a given prime» published in Monatsh. Math.

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María José Felipe, Lev S. Kazarin, Ana Martínez-Pastor, and Víctor Sotomayor.
On products of groups and indices not divisible by a given prime.
Comm. Algebra, 193(4):811–827, 2020.

doi:10.1007/s00605-020-01446-z

Abstract

Let the group G = AB be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each p-regular element of prime power order x ∈ A ∪ B if and only if G is p-decomposable, i.e. G=Op(G) × Op’(G).

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

Keywords: Finite groups; products of groups; conjugacy classes; p-structure; prime graph; almost simple groups

Paper «On large orbits of actions of finite soluble groups: applications» published in Recent advances in pure and applied mathematics

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Adolfo Ballester-Bolinches, Ramon Esteban-Romero, and H. Meng.
On large orbits of actions of finite soluble groups: applications.
Recent advances in pure and applied mathematics. Based on contributions presented at the Second Joint Meeting Spain-Brazil in Mathematics, Cádiz, Spain, December 11–14, 2018, pages 105–113. Cham: Springer, 2020.

doi:10.1007/978-3-030-41321-7_8

Abstract

The main aim of this survey paper is to present two orbit theorems and to show how to apply them to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite soluble groups. We also show how to apply them to solve questions about intersections of some conjugacy families of subgroups of finite soluble groups.

2020 Mathematics Subject Classification: 20C15, 20D10, 20D20, 20D45

Keywords: finite groups, soluble groups, linear groups, regular orbits, formations, prefrattini subgroups, system normalisers

Paper «The number of maximal subgroups and probabilistic generation of finite groups» published in Int. J. Group Theory

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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Paz Jiménez-Seral, Hangyang Meng.
The number of maximal subgroups and probabilistic generation of finite groups.
Int. J. Group Theory, 9(1):31–42, 2020.

doi:10.22108/ijgt.2019.114469.1521

Abstract

In this survey we present some significant bounds for the‎ ‎number of maximal subgroups of a given index of a finite group‎. ‎As a‎ ‎consequence‎, ‎new bounds for the number of random‎ ‎generators needed to generate a finite d-generated group with high‎ ‎probability which are significantly tighter than the ones obtained in‎ ‎the paper of Jaikin-Zapirain and Pyber (Random generation of finite‎ ‎and profinite groups and group enumeration‎, Ann. Math.‎, 183 (2011) 769–814) are obtained‎. ‎The results of‎ ‎Jaikin-Zapirain and Pyber‎, ‎as well as other results of Lubotzky‎, ‎Detomi‎, ‎and Lucchini‎, ‎appear as particular cases of our theorems‎.

2020 Mathematics Subject Classification: 20P05

Keywords: finite group; maximal subgroup; probabilistic generation; primitive group

Paper «On σ-subnormality criteria in finite σ-soluble groups» published in Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM

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A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and V. Pérez-Calabuig.
On σ-subnormality criteria in finite σ-soluble groups.
Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114(2):Paper No. 94, 9, 2020.

doi:10.1007/s00009-019-1444-5

Abstract

Let σ = {σi : i ∈ I} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σ-subnormal in G if there is a chain of subgroups X = X0 ⊆ X1 ⊆⋯⊆ Xn = G where for every j=1,…,n the subgroup Xj-1 is normal in Xj or Xj/CoreXj(Xj-1) is a σi-group for some i ∈ I. In the special case that σ is the partition of ℙ into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality. In this paper some σ-subnormality criteria for subgroups of σ-soluble groups, or groups in which every chief factor is a σi-group, for some i ∈ I, are showed.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: finite group; σ-solubility; σ-nilpotency; σ-subnormal subgroup; factorised group

Paper «On factorised finite groups» published in Mediterr. J. Math.

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A. Ballester-Bolinches, Y. Li, M. C. Pedraza-Aguilera, Ning Su.
On factorised finite groups.
Mediterr. J. Math., 17(2):Paper No. 65, 7, 2020.

doi:10.1007/s00009-020-1500-1

Abstract

A subgroup H of a finite group G is called ℙ-subnormal in G if either H = G or it is connected to G by a chain of subgroups of prime indices. In this paper, some structural results of finite groups which are factorised as the product of two ℙ-subnormal subgroups is showed.

2020 Mathematics Subject Classification: 20D10, 20D25

Keywords: finite group; factorised group; w-supersoluble group; ℙ-subnormal subgroup

Defensa tesi doctoral Hangyang Meng 01/07/2019 12.30

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Jul ’19
1
12:30

El proper dilluns 1 de juliol de 2019, a les 12.30, a la sala de graus «Manuel Valdivia» de la Facultat de Ciències Matemàtiques de la Universitat de València es durà a terme la defensa de la tesi doctoral de Hangyang Meng amb títol

Regular orbits of actions of finite soluble groups. Applications

dirigida per Adolfo Ballester Bolinches.
Esteu tots convidats.

Paper «On products of generalised supersoluble finite groups» published in Mediterr. J. Math.

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A. Ballester-Bolinches, J. Cossey, H. Meng, M. C. Pedraza-Aguilera.
On products of generalised supersoluble finite groups
Ann. Mat. Pura Appl. (4), 198(3):811–819, 2019.

doi:10.1007/s10231-018-0800-6

Abstract

In this paper, mutually sn-permutable subgroups of groups belonging to a class of generalised supersoluble groups are studied. Some analogs of known theorems on mutually sn-permutable products are established.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group, supersoluble group, Sn-permutability, factorisation