Paper «On large orbits of subgroups of linear groups» published in Trans. Amer. Math. Soc.

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H. Meng, A. Ballester-Bolinches y R. Esteban-Romero.
On large orbits of subgroups of linear groups.
Trans. Amer. Math. Soc., 372(4):2589-2612, 2019.

doi:10.1090/tran/7639

Abstract

The main aim of this paper is to prove an orbit theorem and to apply it 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 solvable groups.

2010 Mathematics Subject Classification: 20C15, 20D20, 20D45

Keywords: Finite groups, solvable groups, linear groups, regular orbits, representations of groups

Paper «Left braces and the quantum Yang-Baxter equation» published in Proc. Edinburgh Math. Soc.

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H. Meng, A. Ballester-Bolinches y R. Esteban-Romero.
Left braces and the quantum Yang-Baxter equation.
Proc. Edinburgh Math. Soc., 62(2):595–608, 2019.

doi:10.1017/S0013091518000664

Abstract

Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.

2010 Mathematics Subject Classification: 26T25, 20F16

Keywords: p-nilpotent group, braces, Yang-Baxter equation

Paper «Zeros of irreducible characters in factorized groups» published in Ann. Mat. Pura Appl. (4)

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M. J. Felipe, A. Martínez-Pastor, V. M. Ortiz-Sotomayor.
Zeros of irreducible characters in factorised groups.
Ann. Mat. Pura Appl. (4), 198(1):129–142, 2019.

doi:10.1007/s10231-018-0765-5

Abstract

An element g of a finite group G is said to be vanishing in G if there exists an irreducible character χ of G such that χ(g) = 0; in this case, g is also called a zero of G. The aim of this paper is to obtain structural properties of a factorised group G = AB when we impose some conditions on prime power order elements gAB which are (non-)vanishing in G.

2010 Mathematics Subject Classification: 20D40, 20C15, 20E45

Keywords: Finite groups, products of groups, irreducible characters, conjugacy classes, vanishing elements

Paper «On mutually permutable products of finite groups» published in Rend. Lincei. Mat. Appl.

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Adolfo Ballester-Bolinches, Yangming Li, Mari Carmen Pedraza-Aguilera.

On mutually permutable products of finite groups

Rend. Lincei. Mat. Appl., 29 (4):711-719, 2018

doi:10.4171/RLM/830

Abstract

The main purpose of this paper is to study mutually permutable products G=AB in which the subgroups of prime order p and cyclic of order 4 (if p=2) of the largest normal subgroup of G contained in A \cap B are well situated in G. Our results confirm once again the important role of the intersection of the factors in the structural study of mutually permutable products.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group, Sylow permutability, weakly s-supplementation, factorisation, saturated formation.

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 «A note on normal complements for finite groups» published in Bull. Austral. Math. Soc.

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Ning Su, Adolfo Ballester-Bolinches, Hangyang Meng.

A note on normal complements for finite groups

Bull. Austral. Math. Soc., 98 (1):109-112, 2018

doi:10.1017/S0004972718000151

Abstract

Assume that G is a finite group and H is a 2-nilpotent Sylow tower Hall subgroup of G such that if x and y are G-conjugate elements of H ∩ G0 of prime order or order 4, then x and y are H-conjugate. We prove hat there exists a normal subgroup N of G such that G = HN and H ∩ N = 1.

2010 Mathematics Subject Classification: primary 20D20; secondary 20D10

Keywords: finite group, conjugation, Hall subgroup, normal complement.

Paper «On metacyclic subgroups of finite groups» published in Internat. J. Group Th.

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A. Ballester-Bolinches

On metacyclic subgroups of finite groups

Internat. J. Group Theory, 7(2):25-29, 2018

http://dx.doi.org/10.22108/ijgt.2017.21480

Abstract

The aim of this survey article is to present some structural results about of groups whose Sylow p-subgroups are metacylic (p a prime). A complete characterisation of non-nilpotent groups whose 2-generator subgroups are metacyclic is also presented.

2010 Mathematics Subject Classification: Primary: 20D20; Secondary: 20E25.

Keywords: finite group, Sylow subgroups, metacyclic groups, 2-generator groups.

Paper «On finite groups with square-free conjugacy class sizes» published in Int. J. Group Theory

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María José Felipe, Ana Martínez-Pastor, Víctor-Manuel Ortiz-Sotomayor

On finite groups with square-free conjugacy class sizes

Int. J. Group Th., 7 (2):17-24, 2018

doi:10.22108/ijgt.2017.21475

Abstract

We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.

2010 Mathematics Subject Classification: Primary: 20E45. Secondary: 20D40, 20D10, 20D20

Keywords: finite group, conjugacy class, factorised group

Paper «Some characterisations of groups in which normality is transitive relation by means of subgroup embedding properties» published in Int. J. Group Theory

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Ramón Esteban-Romero and Giovanni Vincenzi.

Some characterisations of groups in which normality is transitive relation by means of subgroup embedding properties

Int. J. Group Th., 7 (2):9-16, 2018

doi:10.22108/ijgt.2017.21214

Abstract

In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in certain universes of groups with some finiteness properties.

2010 Mathematics Subject Classification: Primary: 20D10. Secondary: 20D30, 20D35, 20F19, 20F24

Keywords: group, subgroup embedding property, T-group, FC∗-group, group without infinite simple sections