Paper «On finite involutive Yang-Baxter groups» published in Proc. Amer. Math. Soc.

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H. Meng, A. Ballester-Bolinches, R. Esteban-Romero, and N. Fuster-Corral.
On finite involutive Yang-Baxter groups.
Proc. Amer. Math. Soc., 149(2):793–804, 2021.

doi:10.1090/proc/15283

Abstract

A group G is said to be an involutive Yang-Baxter group, or simply an IYB-group, if it is isomorphic to the permutation group of an involutive, nondegenerate set-theoretic solution of the Yang-Baxter equation. We give new sufficient conditions for a group that can be factorised as a product of two IYB-groups to be an IYB-group. Some earlier results are direct consequences of our main theorem.

2020 Mathematics Subject Classification: Primary 81R50; Secondary 20F29, 20B35, 20F16, 20C05, 16S34, 16T25

Keywords: Finite left brace, Yang-Baxter equation, involutive nondegenerate solutions, involutive Yang-Baxter group

Paper «On finite p-groups of supersoluble type» published in J. Algebra

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A. Ballester-Bolinches, R. Esteban-Romero, H. Meng, and N. Su.
On finite p-groups of supersoluble type.
J. Algebra, 567:1–10, 2021.

doi:10.1016/j.jalgebra.2020.08.025

Abstract

A finite p-group S is said to be of supersoluble type if every fusion system over S is supersoluble. The main aim of this paper is to characterise the finite p-groups of supersoluble type. Abelian and metacyclic p-groups of supersoluble type are completely described. Furthermore, we show that the Sylow p-subgroups of supersoluble type of a finite simple group must be cyclic.

2020 Mathematics Subject Classification: 20D20; 20D15; 20D05

Keywords: finite group; fusion system; supersolubility

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 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 : iI} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σsubnormal in G if there is chain of subgroups X = X0X1 ⊆⋯⊆ Xn = G with Xj-1 normal in Xj or Xi/CoreXi(Xi-1) is a σ-group for some iI, 1 ≤ jn. 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 gAB , 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 «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

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

Paper «The abelian kernel of an inverse semigroup» published in Mathematics

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Adolfo Ballester-Bolinches, Vicent Pérez-Calabuig.
The abelian kernel of an inverse semigroup.
Mathematics, 8(8):1219 (12 pages), 2020.

doi:10.3390/math8081219

Abstract

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of abelian groups with decidable membership. In this paper, we used a completely different approach to complete these results by giving an exact description of the abelian kernel of an inverse semigroup. An abelian group that gives this abelian kernel was also constructed.

2020 Mathematics Subject Classification: 20M10, 20M17

Keywords: finite semigroup; abelian kernels; profinite topologies; partial automorphisms; extension problem

Paper «On large orbits of supersoluble subgroups of linear groups» published in J. Lond. Math. Soc. (2)

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H. Meng, A. Ballester-Bolinches, and R. Esteban-Romero.
On large orbits of supersoluble subgroups of linear groups.
J. Lond. Math. Soc. (2), 101(2):490–504, 2020.

doi:10.1112/jlms.12266

Abstract

We prove that if G is a finite soluble group, V is a finite faithful completely reducible G-module, and H is a supersoluble subgroup of G, then H has at least one regular orbit on VV.

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

Keywords: linear group, regular orbit, supersoluble group

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