Paper «On the Kegel–Wielandt σ‐problem for binary partitions» published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. F. Kamornikov, V. N. Tyutyanov
On the Kegel–Wielandt σ‐problem for binary partitions.
Ann. Mat. Pura Appl., 201:443-451, 2022.

doi: 10.1007/s10231-021-01123-4

Abstract:

Let σ={σ_i: 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= X_0⊆ X_1⊆⋯⊆ X_n= G where, for every i= 1,…, n, the subgroup X_{i− 1} normal in X_ i or X_ i/Core_{X_i} (X_{i− 1}) is a σ_j-group for some j∈ 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. A finite group G is σ-complete if G possesses at least one Hall σ i -subgroup for every i ∈ I , and a subgroup H of G is said to be σ_i-subnormal in G if H ∩ S is a Hall σ_i-subgroup of H for any Hall σ_i-subgroup S of G. Skiba proposes in the Kourovka Notebook the following problem (Question 19.86), that is called the Kegel–Wielandt σ-problem: Is it true that a subgroup H of a σ-complete group G is σ-subnormal in G if H is σ_i-subnormal in G for all i ∈ I? The main goal of this paper is to solve the Kegel–Wielandt σ-problem for binary partitions.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: Finite group; Hall subgroup; σ-subnormal subgroup; factorised group

Paper «On σ-subnormality criteria in finite groups» published in J. Pure Appl. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, X. Yi.
On σ-subnormality criteria in finite groups.
J. Pure Appl. Algebra, 226(2):106822, 2022.

doi: 10.1016/j.jpaa.2021.106822

Abstract:

Let σ={σ_i: i∈ I} be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called σ-subnormal in G if there is a chain of subgroups H= H_0⊆ H_1⊆⋯⊆ H_n= G where, for every i= 1,…, n, H_{i− 1} normal in H i or H i/Core_{H_i} (H_{i− 1}) is a σ_j-group for some j∈ 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 finite groups are studied.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: finite group, σ-nilpotency, σ-subnormal subgroup.

Paper «Generalised mutually permutable products and saturated formations» published in J. Algebra

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A. Ballester-Bolinches, S. Y. Madanha, M. C. Pedraza-Aguilera.
Generalised mutually permutable products and saturated formations.
J. Algebra, 595:434-443, 2022.

doi: 10.1016/j.jalgebra.2021.12.027

Abstract:

We say that a group G = AB is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing AB and B permutes with every subgroup of A containing AB. We prove that some known results for mutually permutable products remain true for weakly mutually permutable ones. Moreover, if G‘ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, we show that G^F = A^FB^F, where is F a saturated formation containing U, the class of supersoluble groups. This generalises the corresponding result on mutually permutable products.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: weakly mutually permutable products, saturated formations, residuals

Paper «Nilpotent length and system permutability» published in J. Algebra

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Rex Darl, Arnold D. Feldman, M. D. Pérez-Ramos.
Nilpotent length and system permutability.
J. Algebra, 589:287-322, 2022.

Abstract:

If C is a class of groups, a C-injector of a finite group G is a subgroup V of G with the property that VK is a C-maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschütz and B. Hartley states the existence and conjugacy of F-injectors in finite soluble groups for Fitting classes F. We shall show that for groups of nilpotent length at most 4, F-injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: Fitting soluble group, Fitting class, injector, system permutability.

Paper «Large characteristically simple sections of finite groups» published in Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat. (RACSAM)

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A. Ballester-Bolinches, R. Esteban-Romero, P. Jiménez-Seral
Large characteristically simple sections of finite groups.
Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat. (RACSAM), 116, Article number 41, 2022.

doi: 10.1007/s13398-021-01188-z

Abstract:

In this paper we prove that if G is a group for which there are k non-Frattini chief factors isomorphic to a characteristically simple group A, then G has a normal section C/R that is the direct product of k minimal normal subgroups of G/R isomorphic to A. This is a significant extension of the notion of crown for isomorphic chief factors.

2020 Mathematics Subject Classification: 20E34, 20E28, 20D10, 20P05.

Keywords: finite group, maximal subgroup, probabilistic generation, primitive group, crown.

Paper «A positive extension of Eilenberg’s variety theorem for non-regular languages» published in Appl. Algebra Eng. Commun. Comp.

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A. Cano, J. Cantero, A. Martínez-Pastor.
A positive extension of Eilenberg’s variety theorem for non-regular languages.
Appl. Algebra Eng. Commun. Comp., 32:553-573, 2021.

doi: 10.1007/s00200-020-00414-2

Abstract:

In this paper we go further with the study initiated by Behle, Krebs and Reifferscheid (in: Proceedings CAI 2011, Lecture Notes in Computer Science, vol 6742, pp 97–114, 2011), who gave an Eilenberg-type theorem for non-regular languages via typed monoids. We provide a new extension of that result, inspired by the one carried out by Pin in the regular case in 1995, who considered classes of languages not necessarily closed under complement. We introduce the so-called positively typed monoids, and give a correspondence between varieties of such algebraic structures and positive varieties of possibly non-regular languages. We also prove a similar result for classes of languages with weaker closure properties.

2020 Mathematics Subject Classification: 68Q70, 68Q45, 20M07, 20M35.

Keywords: monoids, varieties, formal languages.

Paper «A Note on a Paper of Aivazidis, Safonova and Skiba» published in Mediterr. J. Math.

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M. M. Al-Shomrani, Adolfo Ballester-Bolinches, A. A. Heliel.
A Note on a Paper of Aivazidis, Safonova and Skiba.
Mediterr. J. Math, 18: Article number 213, 2021.

doi: 10.1007/s00009-021-01872-9

Abstract:

The main result of this paper states that if F is a subgroup-closed saturated formation of full characteristic, then the F-residual of a K-F-subnormal subgroup S of a finite group G is a large subgroup of G provided that the F-hypercentre of every subgroup X of G containing S is contained in the F-residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: finite group, saturated formation, K-F-subnormal subgroup.

Paper «The Structure Group and the Permutation Group of a Set-Theoretic Solution of the Quantum Yang–Baxter Equation» published in Mediterr. J. Math.

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A. Ballester-Bolinches, R. Esteban-Romero, N. Fuster-Corral, H. Meng.
The Structure Group and the Permutation Group of a Set-Theoretic Solution of the Quantum Yang–Baxter Equation.
Mediterr. J. Math, 18: Article number 145, 2021.

doi: 10.1007/s00009-021-01793-7

Abstract:

We describe the left brace structure of the structure group and the permutation group associated with an involutive, non-degenerate set-theoretic solution of the quantum Yang–Baxter equation using the Cayley graph of its permutation group with respect to its natural generating system. We use our descriptions of the additions in both braces to obtain new properties of the structure and the permutation groups and to recover some known properties of these groups in a more transparent way.

2020 Mathematics Subject Classification: 16T25, 05C25, 20F05, 20F65

Keywords: left brace, Yang-Baxter equation, Cayley graph, structure group.

Paper «Thompson-like characterization of solubility for products of finite groups» published in Ann. Mat. Pura Appl. (4)

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P. Hauck, L. S. Kazarin, A. Martínez-Pastor, and M. D. Pérez-Ramos.
Thompson-like characterization of solubility for products of finite groups.
Ann. Mat. Pura Appl. (4), 200(1):337–362, 2021.

doi:10.1007/s10231-020-00998-z

Abstract

A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson’s theorem from the perspective of factorized groups. More precisely, we study finite groups G = AB with subgroups A, B such that ⟨a, b⟩ is soluble for all aA and bB. In this 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. Our Main Theorem states that G = AB is S-connected if and only if [A, B] is soluble. In the course of the proof, we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.

2020 Mathematics Subject Classification: 20D40, 20D10

Keywords: Solubility, products of subgroups, two-generated subgroups, S-connection, almost simple groups, independent primes

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