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

Talk «Thompson-like characterization of solubility for products of groups» at 2020 Zassenhaus Groups and Friends Conference

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May ’20
29
15:55

María Dolores Pérez Ramos will give the talk entitled

Thompson-like characterization of solubility for products of groups

at the 2020 Zassenhaus Groups and Friends Conference online on 29th May 2020 at 15.55. The link for the talk and its recording appear on http://www2.math.binghamton.edu/p/zassenhaus/zassenhaus_2020/home.

Abstract

A remarkable result of Thompson states that a finite group is soluble if
and only if its two-generated subgroups are soluble. This result has been
sharply generalized, and it is in the core of a wide area of study in the theory
of groups, aiming for global properties of groups from local properties of two-
generated (or more generally, n-generated) subgroups. We report about an
extension of Thompson’s theorem from the perspective of factorized groups.
We prove that for a finite group G = AB, with A, B subgroups of G, if ha, bi
is soluble for all a ∈ A and all b ∈ B, then [A, B] is soluble. In that case, the
group G is said to be an S-connected product of the subgroups A and B, for
the class S of all finite soluble groups. As an application, deep results about
connected products of finite soluble groups, for other relevant classes of
groups, are extended to the finite universe. Collaboration with M. P. Gállego (U.
Zaragoza, Spain), P. Hauck (U. Tübingen, Germany), L. Kazarin (U. Yaroslavl,
Russia), A. Martı́nez-Pastor (U. Politècnica de València, Spain) .

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 V ⊕ V.

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

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 «Congruence-based proofs of the recognizability theorems for free many-sorted algebras» published in J. Logic Comput.

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J. Climent Vidal and E. Cosme Llópez.
Congruence-based proofs of the recognizability theorems for free many-sorted algebras.
J. Logic Comput., 30(2):561–633, 2020.

doi:10.1093/logcom/exz032

Abstract

We generalize several recognizability theorems for free single-sorted algebras to free many-sorted algebras and provide, in a uniform way and without using either regular tree grammars or tree automata, purely algebraic proofs of them based on congruences.

Keywords: free many-sorted algebra, recognizability, congruence

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

Visita i xarrada professor Jean-Éric Pin 24/02/2020

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Feb ’20
24
12:00

Jean-Éric PinBenvolgudes companyes, benvolguts companys,

El professor Jean-Éric Pin (IRIF, CNRS i Université Paris-Diderot) ens visitarà el proper dilluns 24 de febrer i impartirà la xarrada

«Formations of monoids»

el proper dilluns 24 de febrer a les 12.00 a l’aula 1.5 de la Facultat de Matemàtiques.

Ben cordialment,

Ramon

Paper «An elementary proof of a theorem of Graham on finite semigroups» published in Mathematics

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Adolfo Ballester-Bolinches and Vicent Pérez-Calabuig
An elementary proof of a theorem of Graham on finite semigroups.
Mathematics, 8(1):105 (5 pages), 2020.

doi:10.3390/math8010105

Abstract

The purpose of this note is to give a very elementary proof of a theorem of Graham that provides a structural description of finite 0-simple semigroups and its idempotent-generated subsemigroups.

2010 Mathematics Subject Classification: 20M10, 20M17

Keywords: finite semigroup; regular semigroup; 0-simple semigroup