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 «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 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.

Charla de Ana Martínez Pastor en «Finite groups in Valencia», 30/03/2021, 17.40

Mar ’21
30
17:40

El próximo 30 de marzo de 2021 de 17.40 a 18.25, Ana Martínez Pastor pronunciará la charla «Hall-like theorems in products of π-decomposable groups» en el congreso Finite groups in Valencia. Más información en

https://sites.google.com/view/finite-groups-seminar2021/

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

Paper «On finite p-groups of supersoluble type» published in J. Algebra

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A. Ballester-Bolinches, R. Esteban-Romero, H. Meng, N. Su
On certain products of permutable subgroups.
J. Algebra, 567, 1-10.

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

Paper «Products of groups and class sizes of π-elements» published in Mediterr. J. Math.

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M. J. Felipe, A. Martínez-Pastor, V. M. Ortiz-Sotomayor.
Products of groups and class sizes of π-elements.
Mediterr. J. Math., 17(1):Paper No. 15, 20, 2020.

doi:10.1007/s00009-019-1444-5

Abstract

We provide structural criteria for some finite factorised groups G=AB when the conjugacy class sizes in G of certain π-elements in AB are either π-numbers or π′-numbers, for a set of primes π. In particular, we extend for products of groups some earlier results.

2020 Mathematics Subject Classification: 20D10, 20D40, 20E45, 20D20

Keywords: finite group; products of groups; conjugacy classes, π-structure

Paper «On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups» published in Mathematics

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Abd El-Rahman Heliel, Mohammed Al-Shomrani, Adolfo Ballester-Bolinches.
On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups.
Mathematics, 8(12):2165 (4 pages), 2020.

doi:10.3390/math8122165

Abstract

Let σ={σi:iI} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by (G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by (G). If F is a subgroup-closed saturated formation, we define the σ-F-length (G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then (A,F)=(G,F)−i for some i∈{0,1,2}.

Keywords: finite group; σ-solubility; σ-nilpotency; σ-nilpotent length

Paper «The Dπ-property on products of π-decomposable groups» published in Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM

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L. S. Kazarin, A. Martínez-Pastor, and M. D. Pérez-Ramos.
The Dπ-property on products of π-decomposable groups.
Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115(1):Paper No. 13, 18, 2021.

doi:10.1007/s13398-020-00950-z

Abstract

The aim of this paper is to prove the following result: Let π be a set of odd primes. If the group G = AB is the product of two π-decomposable subgroups A = Aπ × Aπ′ and B = Bπ × Bπ′, then G has a unique conjugacy class of Hall π-subgroups, and any π-subgroup is contained in a Hall π-subgroup (i.e. G satisfies property Dπ).

2020 Mathematics Subject Classification: 20D40; 20D20; 20E32

Keywords: finite groups; product of subgroups; π-structure; simple groups