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

Paper «Groups whose subgroups satisfy the weak subnormalizer condition» published in Beitr. Algebra Geom.

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R. Esteban Romero, F. de Giovanni, A. Russo.
Groups whose subgroups satisfy the weak subnormalizer condition
Beitr. Algebra Geom., 60(4):645–656, 2019.

doi:10.1007/s13366-019-00448-9

Abstract

A subgroup X of a group G is said to satisfy the weak subnormalizer condition if NG(Y)NG(X) for each non-normal subgroup Y of G such that X≤Y≤NG(X). The behaviour of generalized soluble groups whose (cyclic) subgroups satisfy the weak subnormalizer condition is investigated.

2010 Mathematics Subject Classification: 20E15, 20F16

Keywords: Weak subnormalizer condition, T-group, weakly radical group

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 «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 «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

Paper “Finite groups with all minimal subgroups solitary” published in J. Algebra Appl.

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R. Esteban-Romero and Orieta Liriano.

Finite groups with all minimal subgroups solitary.

J. Algebra Appl., 15(8):1650140, 9, 2016

https://doi.org/10.1142/S0219498816501401

Abstract 

We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order.

2010 Mathematical Subject Classification: 20D10, 20D30

Keywords: Finite group; solitary subgroup; minimal subgroup

Paper “Some Local Properties Defining T₀-Groups and Related Classes of Groups” published in Publ. Mat.

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A. Ballester-Bolinches, J. C. Beidleman, R. Esteban-Romero, and M. F. Ragland

Some local properties defining T0-groups and related classes of groups

Publ. Mat., 60(1):265–272, 2016

http://projecteuclid.org/euclid.pm/1450818490

Abstract

We call G a Hall_χ-group if there exists a normal nilpotent subgroup N of G for which G/N is an χ-group. We call G a T-group provided G/Φ(G) is a T-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define Hall_χ-groups and T-groups where χ{T, PT, PST}; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.

2010 Mathematical Subject Classification: 20D10, 20D20, 20D35

Keywords: Subnormal subgroup, T-group, PST-group, finite solvable group