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 «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 the Prüfer rank of mutually permutable products of abelian groups» published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, J. Cossey, H. Meng, M. C. Pedraza-Aguilera.
On the Prüfer rank of mutually permutable products of abelian groups
Ann. Mat. Pura Appl. (4), 198(3):811–819, 2019.

doi:10.1007/s10231-018-0800-6

Abstract

A group G has finite (or Prüfer or special) rank if every finitely generated subgroup of G can be generated by r elements and r is the least integer with this property. The aim of this paper is to prove the following result: assume that G=AB is a group which is the mutually permutable product of the abelian subgroups A and B of Prüfer ranks r and s, respectively. If G is locally finite, then the Prüfer rank of G is at most r+s+3. If G is an arbitrary group, then the Prüfer rank of G is at most r+s+4.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Abelian group · Soluble group · Polycyclic group · Rank · Factorisations

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 locally finite groups whose subgroups of infinite rank have some permutable property” published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. Camp-Mora, M. R. Dixon, R. Ialenti, and F. Spagnuolo

On locally finite groups whose subgroups of infinite rank have some permutable property

Ann. Mat. Pura Appl. (4), 196(5):1855–1862, 2017

https://doi.org/10.1007/s10231-017-0642-7

Abstract

In this paper, we study the behavior of locally finite groups of infinite rank whose proper subgroups of infinite rank have one of the three following properties, which are generalizations of permutability: S-permutability, semipermutability and S-semipermutability. In particular, it is proved that if G is a locally finite group of infinite rank whose proper subgroups of infinite rank are S-permutable (resp. semipermutable), then G is locally nilpotent (resp. all subgroups are semipermutable). For locally finite groups whose proper subgroups of infinite rank are S-semipermutable, the same statement can be proved only for groups with min-p for every prime p. A counterexample is given for the general case.

2010 Mathematical Subject Classification: 20F19, 20F50

Keywords: Locally finite group, Section p-rank, Section rank, Special rank, Permutable, Sylow permutable, Semipermutable, S-semipermutable

 

Paper «On a class of generalised Schmidt groups» published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, R. Esteban-Romero, Qinhui Jiang, and Xianhua Li

On a class of generalised Schmidt groups

Ann. Mat. Pura Appl. (4), 194(1) (2015), 77–86

http://dx.doi.org/10.1007/s10231-013-0365-3

Abstract

In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An A_5-free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained.

2010 Mathematics subject classification: 20D05; 20D10; 20F16

Keywords: finite groups; nilpotent groups; maximal subgroups

Paper «Prefactorized subgroups in pairwise mutually permutable products» published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, M. C. Pedraza-Aguilera

Prefactorized subgroups in pairwise mutually permutable subgroups

Ann. Math .Pura Appl., 192(6), 1043-1057 (2013)

http://dx.doi.org/10.1007/s10231-012-0257-y

Abstract

We continue here our study of pairwise mutually and pairwise totally permutable products. We are looking for subgroups of the product in which the given factorization induces a factorization of the subgroup. In the case of soluble groups, it is shown that a prefactorized Carter subgroup and a prefactorized system normalizer exist. A less stringent property have F-residual, F-projector and F-normalizer for any saturated formation F including the supersoluble groups.

MSC: 20D10, 20D20

Keywords: Finite group, Permutability, Factorization, Saturated formation.

Paper «On a class of generalised Schmdit groups» published in Ann. Mat. Pura Appl.

The paper

A. Ballester-Bolinches, R. Esteban-Romero, Qinhui Jiang, Xianhua Li

On a class of generalised Schmidt groups

will be published in Annali di Matematica Pura ed Applicata. It is available through

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A. Ballester-Bolinches, R. Esteban-Romero, Qinhui Jiang, Xianhua Li

On a class of generalised Schmidt groups

será publicado en Annali di Matematica Pura ed Applicata. Está disponible en

http://dx.doi.org/10.1007/s10231-013-0365-3
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L’article

A. Ballester-Bolinches, R. Esteban-Romero, Qinhui Jiang, Xianhua Li

On a class of generalised Schmidt groups

serà publicat en Annali di Matematica Pura ed Applicata. Està disponible en

http://dx.doi.org/10.1007/s10231-013-0365-3

(vegeu resum més avall). Informarem sobre els detalls de la seua publicació.

Abstract: In this paper families of non-nilpotent subgroups covering the non-nilpotent part
of a finite group are considered. An A_5-free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained.

Keywords:  Finite groups · Nilpotent groups · Maximal subgroups
Mathematics Subject Classification (2010):  20D05 · 20D10 · 20F16