# 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.

# Talk «Carter and Gaschütz theories revisited» by María Dolores Pérez-Ramos at Ischia Online Group Theory Conference (GOThIC), 09/12/2021, 18.00

 Dic ’21 9 18:00

The Organizing Committee of the  Ischia Online Group Theory Conference(GOThIC) is inviting you to a scheduled Zoom meeting.
– The TIME OF THE TALK is 18:00 CET (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 ANDSURNAME, or  close to it, so  that the organisers can recognise you and let you in.
TOPIC: GOThIC – Ischia Online Group Theory Conference –  (https://sites.google.com/unisa.it/e-igt2020/).
TIME: Thursday December 9th, 2021 18:00 CET (UTC+1).
COFFEE BREAK: The  talk will start at 18:00 CET.  The conference roomwill open at 17:45 CET for a drink – Bring Your Own appropriate drink – biscuits appreciated – and join us for some smalltalk before the event.
SPEAKER: Maria Dolores Pérez -Ramos (Universitat de València)
TITLE: Carter and Gaschütz theories revisited

ABSTRACT: Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. We present an extension of these facts to π-separable groups, for sets of primes π, by proving the existence of a conjugacy class of subgroups in π-separable groups, which specialize to Carter subgroups within the universe of soluble groups.

The approach runs parallel to the extension of Hall theory from soluble to π-separable groups by Cunihin, regarding existence and properties of Hall subgroups. Our Carter-like subgroups enable an extension of the existence and conjugacy of injectors associated to Fitting classes to π-separable groups, in the spirit of the role of Carter subgroups in the theory of soluble groups. This is joint work with M. Arroyo-Jordá, P. Arroyo-Jordá, R. Dark and A.D. Feldman.

# 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

# Talk «Thompson-like characterization of solubility for products of groups» at 2020 Zassenhaus Groups and Friends Conference

 May ’20 29 15:55

María Dolores Pérez Ramos will give the talk entitled

Thompson-like characterization of solubility for products of groups

at the 2020 Zassenhaus Groups and Friends Conference online on 29th May 2020 at 15.55. The link for the talk and its recording appear on http://www2.math.binghamton.edu/p/zassenhaus/zassenhaus_2020/home.

Abstract

A remarkable result of Thompson states that a finite group is soluble if
and only if its two-generated subgroups are soluble. This result has been
sharply generalized, and it is in the core of a wide area of study in the theory
of groups, aiming for global properties of groups from local properties of two-
generated (or more generally, n-generated) subgroups. We report about an
extension of Thompson’s theorem from the perspective of factorized groups.
We prove that for a finite group G = AB, with A, B subgroups of G, if ha, bi
is soluble for all a ∈ A and all b ∈ B, then [A, B] is soluble. In that 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. As an application, deep results about
connected products of finite soluble groups, for other relevant classes of
groups, are extended to the finite universe. Collaboration with M. P. Gállego (U.
Zaragoza, Spain), P. Hauck (U. Tübingen, Germany), L. Kazarin (U. Yaroslavl,
Russia), A. Martı́nez-Pastor (U. Politècnica de València, Spain) .

# 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.

# Paper “2-Engel relations between subgroups” published in J. Algebra

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M. P. Gállego, P. Hauck, and M. D. Pérez-Ramos

2-Engel relations between subgroups

J. Algebra, 447:31–55, 2016

https://doi.org/10.1016/j.jalgebra.2015.08.030

## Abstract

In this paper we study groups G generated by two subgroups A and B such that 〈a, bis nilpotent of class at most 2 for all aA and  bB . A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups.

2010 Mathematical Subject Classification: Primary 20F45; Secondary 20F18, 20D15, 20F12

Keywords: 2-Engel condition, 2-generated subgroups, L-connection, Nilpotency class