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

Talk «Triply factorised groups and skew left braces» at Ischia Online Group Theory Conference (GOThIC) on 19th November, 2020

Nov ’20
19
17:00

The organising committee of the
Ischia Online Group Theory Conference(GOThIC)
is inviting you to a scheduled Zoom meeting.

PLEASE NOTE:

– The TIME OF THE TALK is 17:00 CET = UTC + 1.

– You are welcome to share the Zoom link with other interested
parties, but PLEASE DO NOT POST THE LINK PUBLICLY.

– When joining, please MAKE SURE THAT YOUR NICKNAME
IS YOUR NAME AND SURNAME, or close to it, so that the organisers
can recognise you and let you in

The Ischia Group Theory 2020 Conference
(http://www.dipmat2.unisa.it/ischiagrouptheory/) was planned
for 30 March – 4 April 2020. It has now been postponed.
In the meantime, we are offering a series of online lectures
by leading researchers (https://sites.google.com/unisa.it/e-igt2020/).

TIME: November 19th, 2020 17:00 CET (UTC+1)

COFFEE BREAK: The talk will start at 17:00 CET. The conference room
will open at 16:45 CET for a coffee break
– Bring Your Own tea/coffee mug – biscuits appreciated –
and join us for some smalltalk before the event.

SPEAKER: Ramon Esteban-Romero (Universitat de València)

TITLE: Triply factorised groups and skew left braces

ABSTRACT:

The Yang-Baxter equation is a consistency equation of the statistical mechanics proposed by Yang [6] and Baxter [1] that describes the interaction of many particles in some scattering situations. This equation lays the foundation for the theory of quantum groups and Hopf algebras. During the last years, the study suggested by Drinfeld [2] of the so-called set-theoretic solutions of the Yang-Baxter equation has motivated the appearance of many algebraic structures. Among these structures we find the skew left braces, in troduced by Guarnieri and Vendramin [3] as a generalisation of the structure of left brace defined by Rump [4]. It consists of a set B with two operations + and ·, not necessarily commutative, that give B two structures of group linked by a modified distributive law.

The multiplicative group C = (B, ·) of a skew left brace (B, +, ·) acts on the multiplicative group K = (B, +) by means of an action λ: C −→ Aut(K) given by λ(a)(b) = −a + a · b, for a, b ∈ B. With respect to this action, the identity map δ : C −→ K becomes a derivation or 1-cocycle with respect to λ. In the semidirect product G = [K]C = {(k, c) | k ∈ K, c ∈ C}, there is a diagonal-type subgroup D = {(δ(c), c) | c ∈ C} such that G = KD = CD, K ∩ D = C ∩ D = 1. This approach was presented by Sysak in [5] and motivates the use of techniques of group theory to study skew left braces.

We present in this talk some applications of this approach to obtain some results about skew left braces. These results have been obtained in collaboration with Adolfo Ballester-Bolinches.

Recorded talks: https://sites.google.com/unisa.it/e-igt2020/recorded-talks

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

The following paper has been published:
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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.