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Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.
Assume that
$G$
is a finite group and
$H$
is a 2-nilpotent Sylow tower Hall subgroup of
$G$
such that if
$x$
and
$y$
are
$G$
-conjugate elements of
$H\cap G^{\prime }$
of prime order or order 4, then
$x$
and
$y$
are
$H$
-conjugate. We prove that there exists a normal subgroup
$N$
of
$G$
such that
$G=HN$
and
$H\cap N=1$
.
Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.
Some classes of finitely generated hyperabelian groups defined in terms of semipermutability and S-semipermutability are studied in the paper. The classification of finitely generated hyperabelian groups all of whose finite quotients are PST-groups recently obtained by Robinson is behind our results. An alternative proof of such a classification is also included in the paper.
In this paper we analyse the structure of a finite group of minimal order among the finite non-supersoluble groups possessing a triple factorization by supersoluble subgroups of pairwise relatively prime indices. As an application we obtain some sufficient conditions for a triple factorized group by supersoluble subgroups of pairwise relatively prime indices to be supersoluble. Many results appear as consequences of our analysis.
For a formation
$\mathfrak F$
, a subgroup M of a finite group G is said to be
$\mathfrak F$
-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉
$\mathfrak F$
such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be
$\mathfrak F$
-normal in G if G/CoreG(M) belongs to
$\mathfrak F$
. A subgroup U of a finite group G is called K-
$\mathfrak F$
-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or
$\mathfrak F$
-normal in Ui, for i = 1,2, …, n. We call a finite group G an
$fT_{\mathfrak F}$
-group if every K-
$\mathfrak F$
-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations
$\mathfrak F$
the structure of
$fT_{\mathfrak F}$
-groups. We pay special attention to the
$\mathfrak F$
-pronormal subgroups in this analysis.
A subgroup
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$
of a finite group
$G$
is said to be S-semipermutable in
$G$
if
$H$
permutes with every Sylow
$q$
-subgroup of
$G$
for all primes
$q$
not dividing
$|H |$
. A finite group
$G$
is an MS-group if the maximal subgroups of all the Sylow subgroups of
$G$
are S-semipermutable in
$G$
. The aim of the present paper is to characterise the finite MS-groups.
All groups considered in this paper are finite. A subgroup
$H$
of a group
$G$
is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of
$G$
containing
$H$
as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc.28(1) (2013), 55–62] proved that every primitive subgroup of
$G$
has index a power of a prime if and only if
$G/ \Phi (G)$
is a solvable PST-group. Let
$\mathfrak{X}$
denote the class of groups
$G$
all of whose primitive subgroups have prime power index. It is established here that a group
$G$
is a solvable PST-group if and only if every subgroup of
$G$
is an
$\mathfrak{X}$
-group.
A subgroup A of a group G has the strong cover-avoidance property inG, or A is a strong CAP-subgroup ofG, if A either covers or avoids every chief factor of every subgroup of G containing A. The main aim of the present paper is to analyse the impact of the strong cover and avoidance property of the members of some relevant families of subgroups on the structure of a group.
By
A. Ballester-Bolinches, Departament d'Álgebra, Universitat de Valéncia, Dr. Moliner, 50, 46100 Burjassot (Valéncia), Spain,
Clara Calvo,
R. Esteban-Romero, Departament de Matemática Aplicada-IMPA, Universitat Politécnica de Valéncia, Camí de Vera, s/n, 46022 Valéncia, Spain
By
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de Valéncia, C/ Doctor Moliner 50, 46100 Burjassot (Valéncia), Spain,
John Cossey, Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra, 0200, Australia,
M. C. Pedraza-Aguilera, ETS de Informática Aplicada, Departamento de Matemática Aplicada-IMPA, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain
The study of factorized groups has played an important role in the theory of groups. We can consider so relevant results as the Ito's Theorem about products of abelian groups or the celebrated Theorem of Kegel-Wielandt about the solubility of a product of two nilpotent groups. In the very much special case when the factors are normal and nilpotent, a well-known result due to Fitting shows that the product is nilpotent. Nevertheless it is not true in general that the product of two normal supersoluble subgroups of a group is a supersoluble group. To create intermediate situations it is usual to consider products of groups whose factors satisfy certain relations of permutability. Following Carocca [12] we say that a group G = AB is the mutually permutable product of A and B if A permutes with every subgroup of B and vice versa. If, in addition, every subgroup of A permutes with every subgroup of B, we say that the group G is a totally permutable product of A and B.
In this context, we can consider as seminal the following results of Asaad and Shaalan.
Theorem A (Asaad and Shaalan [2]) (i) Assume that a group G = AB is the mutually permutable product of A and B. Suppose that A and B are supersoluble and that either A, B or G', the derived subgroup of G, is nilpotent. Then G is supersoluble.
By
A. Ballester-Bolinches, Departament d'Álgebra, Universitat de Valéncia, c/ Doctor Moliner 50, 46100 Burjassot (Valencia), Spain,
H. Heineken, Mathematisches Institut, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany,
Tatiana Pedraza, ETS de Informática Aplicada, Departamento de Matemática Aplicada-IMPA, Universidad Politécnica de Valencia, 46022 Valencia, Spain
A group G is said to be a T-group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation in G. The study of this class of groups begins with the publication of a paper of Dedekind in 1896. He characterizes the finite groups in which every subgroup is normal. These groups, called Dedekind groups, are obvious examples of T-groups. The extension of Dedekind's result to infinite groups was proved by Baer in 1933.
Theorem (Dedekind, Baer)All the subgroups of a group G are normal if and only if G is abelian or the direct product of a quaternion group of order 8, an elementary abelian 2-group and an abelian group with all its elements of odd order.
In 1942, E. Best and O. Taussky [5] prove that every finite group with cyclic Sylow subgroups is a T-group. Later G. Zacher characterized soluble finite Tgroups by means of Sylow towers properties (see [12]). However, the decisive result about the structure of T-groups in the finite soluble universe was obtained by Gaschütz in 1957 ([8]).
Theorem (Gaschütz)Let G be a finite soluble group. Then G is a T-group if and only if it has an abelian normal Hall subgroup L of odd order such that G/L is a Dedekind group and the elements of G induce power automorphisms in L.
It is shown in this paper that if is a class of simple groups such that π() = char , the -saturated formation ℌ generated by a finite group cannot be expressed as the Gaschütz product ∘ of two non--saturated formations if ℌ ≠ . It answers some open questions on products of formations. The relation between ω-saturated and -saturated formations is also discussed.
By
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain,
Tatiana Pedraza, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain; This work is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain
By
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain,
John Cossey, Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra, 0200, Australia,
Xiuyun Guo, Department of Mathematics, Shanxi University, Taiyuan, Shanxi 030006, People's Republic of China,
M. C. Pedraza-Aguilera, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain; The work of the first and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain
The well-known fact that a product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that the saturated formation of supersoluble groups need not be closed under the product of normal subgroups. This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties. Baer (see [2]) proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. This result has been generalized by Asaad and Shaalan ([1]) in the following sense: If G is the product of two subgroups H and K such that H permutes with every subgroup of K and K permutes with every subgroup of H, that is, G is the mutually permutable product of H and K, and G', the commutator subgroup of G is nilpotent, then G is supersoluble. Moreover they prove that in the case G = HK such that every subgroup of H permutes with every subgroup of K, that is, G is the totally permutable product of H and K, then if the factors H and K are supersoluble the group G is also supersoluble.
Further studies have been done by several authors within the framework of formation theory.
By
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain,
A. Martínez-Pastor, Escuela Universitaria de Informática, Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain,
M.C. Pedraza-Aguilera, Escuela Universitaria de Informática, Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain,
M.D. Pérez-Ramos, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain; This research has been supported by Proyecto PB 97-0674-C02-02 of DGICYT, Ministerio de Educación y Ciencia, Spain.
By
Manuel J. Alejandre, Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. del Ferrocarril s/n, 03202 Elche, Spain,
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain,
R. Esteban-Romero, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain,
M. C. Pedraza-Aguilera, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain; The work of the second and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain
A group G is said to be a T-group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation. These groups have been widely studied (see [10], [11], or [14]).
A subgroup H of a group G is said to be permutable (or quasinormal) in G if HK = KH for all subgroups K of G. Permutability can be considered thus as a weak form of normality. The study of groups G in which permutability is transitive, that is, H permutable in K and K permutable in G always imply that H is permutable in G, has been a successful field of research in recent years. Such groups are called PT-groups. According to a theorem of Kegel [12, Satz 1], every permutable subgroup of G is subnormal in G. Consequently, PT-groups are exactly those groups in which subnormality and permutability coincide; that is, those groups in which every subnormal subgroup permutes with every other subgroup. Therefore, every T-group is clearly a PT-group.
One could wonder what would happen if we did not require that every subnormal subgroup of a group G permutes with any other subgroup of G, but only with a certain family of its subgroups. In this direction, those groups in which every subnormal subgroup of G permutes with every Sylow p-subgroup of G for each prime p have sometimes been called T*-groups (see [3]) or also (π – q)-groups (see [1]).
Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.