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

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 «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

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 σ-subnormal subgroups of factorised finite groups» published in J. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and X. Yi.
On σ-subnormal subgroups of factorised finite groups.
J. Algebra, 559:195–202, 2020.

doi:10.1016/j.jalgebra.2020.05.002

Abstract

Let σ = {σi : iI} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σsubnormal in G if there is chain of subgroups X = X0X1 ⊆⋯⊆ Xn = G with Xj-1 normal in Xj or Xi/CoreXi(Xi-1) is a σ-group for some iI, 1 ≤ jn. 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.

If a finite soluble group G = AB is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is σ-subnormal in 〈X, Xg〉 for all gAB , we prove that X is σ-subnormal in G. This is an extension of a subnormality criteria due to Maier and Sidki and Casolo.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; Soluble group; σ-Subnormal subgroup; σ-Nilpotency; Factorised group

Paper «Products of finite connected subgroups» published in Mathematics

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María Pilar Gállego, Peter Hauck, Lev S. Kazarin, Ana Martínez-Pastor, and María Dolores Pérez-Ramos.
Products of finite connected subgroups.
Mathematics, 18(9):1498 (8 pages), 2020.

doi:10.3390/math8091498

Abstract

For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if ⟨a,b⟩∈L for all aA and bB. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

2020 Mathematics Subject Classification: 20D40, 20E45, 20D20, 20D60

Keywords: finite groups; products of subgroups; two-generated subgroups; L-connection; Fitting classes; Fitting series; formations

Paper «On σ-subnormal closure» published in Comm. Algebra

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M. M. Al-Shomrani, A. A. Heliel, and Adolfo Ballester-Bolinches.
On σ-subnormal closure.
Comm. Algebra, 48(8):3624–3627, 2020.

doi:10.1080/00927872.2020.1742348

Abstract

Let σ = {σi : iI} be a partition of the set ℙ of all prime numbers. A subgroup A of a finite group G is called σsubnormal in G if there is chain of subgroups A = A0A1 ⊆⋯⊆ An = G with Aj-1 normal in Aj or Ai/CoreAi(Ai-1) is a σj-group for some jI, 1 ≤ in. In this paper, the description of the unique smallest σ-subnormal subgroup of a σ-soluble group containing a given subgroup is obtained.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group; σ-soluble group; σ-subnormal subgroup