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 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 : iI} 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 = X0X1 ⊆⋯⊆ 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 iI. 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 iI, are showed.

2020 Mathematics Subject Classification: 20D10, 20D20

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

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

Defensa tesi doctoral Vicent Pérez Calabuig 18/12/2018 12.30

Dic ’18
18
12:30

El proper dimarts 18 de desembre de 2018, 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 Vicent Pérez Calabuig amb títol «A reduction theorem for the generalised Rhodes’ Type II Conjecture» dirigida per Adolfo Ballester Bolinches.
Esteu tots convidats.

 

Paper «On formations of finite groups with the generalized Wielandt property for residuals II» published in J. Algebra Appl.

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A. Ballester-Bolinches, S. F. Kamornikov, V. Pérez-Calabuig

On formations of finite groups with the generalized Wielandt property for residuals II

J. Algebra Appl., 17 (9):1850167 (8 pages), 2018

http://dx.doi.org/10.1142/S0219498818501670

Abstract

A formation F of finite groups has the generalized Wielandt property for residuals, or F is a GWP-formation, if the F-residual of a group generated by two F-subnormal subgroups is the subgroup generated by their F-residuals. The main result of this paper describes a large family of GWP-formations to further the transparence of this kind of formations, and it can be regarded as a natural step toward the solution of the classification problem.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; formation; residual; subnormality.

Paper «A note on the rational canonical form of an endomorphism of a vector space of finite dimension» published in «Operators and Matrices»

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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Vicente Pérez-Calabuig

A note on the rational canonical form of an endomorphism of a vector space of finite dimension

Operators and Matrices, 12 (3):823-836, 2018

doi:10.7153/oam-2018-12-49

Abstract

In this note, we give an easy algorithm to construct the rational canonical form of a square matrix or an endomorphism h of a finite dimensional vector space which does not depend on either the structure theorem for finitely generated modules over principal ideal domains or matrices over the polynomial ring. The algorithm is based on the construction of an element whose minimum polynomial coincides with the minimum polynomial of the endomorphism and on the fact that the h-invariant subspace generated by such an element admits an h-invariant complement. It is also shown that this element can be easily obtained without the factorisation of a polynomial as a product of irreducible polynomials.

2010 Mathematics Subject Classification: 15A21, 12E05, 12Y05

Keywords: Similarity of matrices, rational canonical form, minimum polynomial, endomorphism.

Paper “On complements of F-residuals of finite groups” published in Comm. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, and V. Pérez-Calabuig

On Complements of F-residuals of finite groups

Comm. Algebra, 45(2):878–882, 2017.

https://doi.org/10.1080/00927872.2016.1175615

Abstract

A formation F of finite groups has the generalized Wielandt property for residuals, or is a GWP-formation, if the F-residual of a group generated by two F-subnormal subgroups is the subgroup generated by their F-residuals. The main aim of the paper is to determine some sufficient conditions for a finite group to split over its F-residual.

2010 Mathematics subject classification: 20D10; 20D20

Keywords: Finite group; formation; residual; subnormality

Paper “On seminormal subgroups of finite groups” published in Rocky Mountain J. Math.

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A. Ballester-Bolinches, J. C. Beidleman, V. Pérez-Calabuig, and M. F. Ragland

On seminormal subgroups of finite groups

Rocky Mountain J. Math., 47(2):419–427, 2017

https://doi.org/10.1216/RMJ-2017-47-2-419

Abstract

All groups considered in this paper are finite. A subgroup H of a group G is said to be seminormal in G if H is normalized by all subgroups K of G such that gcd(|H|,|K|)=1 . We call a group G an MSN-group if the maximal subgroups of all the Sylow subgroups of G are seminormal in G. In this paper, we classify all MSN-groups.

2010 Mathematics Subject Classification: 20D10, 20D15, 20D20

Keywords: Finite group, soluble PST-group,T₀-group, MS-group, MSN-group

 

Paper «On formations of finite groups with the generalised Wielandt property for residuals» published in J. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, and V. Pérez-Calabuig

On formations of finite groups with the generalised Wielandt property for residuals

J. Algebra., 412 (2014), 173–178

http://dx.doi.org/10.1007/s11856-013-0030-y

Abstract

A formation F of finite groups has the generalised Wielandt property for residuals, or F is a GWP-formation, if the F-residual of a group generated by two F-subnormal subgroups is the subgroup generated by their F-residuals. We prove that every GWP-formation is saturated. This is one of the crucial steps in the hunt for a solution of the classification problem.

2010 Mathematics subject classification: 20D10; 20D20

Keywords: finite group; formation; residual; subnormality

Publication data for «Maximal subgroups and PST-groups» in Cent. Eur. Math. J.

Central European Journal of MathematicsWe now have the issue and page numbers for the paper we mentioned in http://permut.blogs.uv.es/2013/03/15/paper-maximal-subgroups-and-pst-groups/.

Adolfo Ballester-Bolinches, James C. Beidleman, Ramón Esteban-Romero, Vicent Pérez-Calabuig

Maximal subgroups and PST-groups

Centr. Eur. J. Math., 11(6), 2013, 1078-1082,

available on http://dx.doi.org/10.2478/s11533-013-0222-z.

Abstract:

A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.

MSC:  20D05, 20D10, 20E15, 20E28, 20F16
Keywords: Finite groups • Permutability • Sylow-permutability • Maximal subgroups • Supersolubility

(c) Versita Sp. z. o. o. and Springer