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 «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

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 “Finite trifactorised groups and $\pi$-decomposability” published in Bull. Austral. Math. Soc.

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L. S. Kazarin, A. Martínez-Pastor, M. D. Pérez-Ramos.

Finite trifactorised groups and $\pi$-decomposability

Bull. Austral. Math. Soc., 97 (2):218-228, 2018

doi:10.1017/S0004972717001034

Abstract

We derive some structural properties of a trifactorised finite group G = AB = AC = BC, where A, B, and C are subgroups of G, provided that A = Aπ × Aπ’ and B = Bπ × Bπ’ are π-decomposable groups, for a set of primes π.

2010 Mathematics Subject Classification: primary 20D40; secondary 20D20

Keywords: finite group, product of subgroups, π-decomposable group, π-structure.