Abelian Kernel Research Articles

Pro-[formula omitted] RAAGs

Let C be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-C group GΓ (pro-C RAAG for short) is the pro-C completion of the right-angled Artin group G(Γ) associated with the finite simplicial graph Γ.In the first part, we describe structural properties of pro-C RAAGs. Among others, we describe the centraliser of an element and show that pro-C RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-p subgroups of pro-C RAAGs are either free pro-p or free abelian pro-p.In the second part, we characterise splittings of pro-C RAAGs in terms of the defining graph. More precisely, we prove that a pro-C RAAG GΓ splits as a non-trivial direct product if and only if Γ is a join and it splits over an abelian pro-C group if and only if a connected component of Γ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-C RAAG, in the sense of Guirardel and Levitt [GL17].

Split epimorphisms and Baer sums of skew left braces

We investigate the split epimorphisms in the categories of digroups and skew left braces. We show that, unlike the category DiGp of digroups, the category SkB of skew left braces is strongly protomodular. From that, we describe the expected Baer sums of exact sequences of skew left braces with abelian kernel.

THE DEVELOPMENT OF A PIEZOELECTRIC DEFECT DETECTION DEVICE THROUGH MATHEMATICAL MODELING APPLIED TO POLYMER COMPOSITE MATERIALS

In this research article, the focus is on examining the strength characteristics of polymer composite materials under various stress states. The investigation into the material dissolution process is conducted in two distinct stages. Initially, the article employs a criterion approach to derive explicit formulas for the latent decay time (incubation period) of materials, considering singular, exponential and Abelian kernels of the damage operator. In the subsequent stage, the classical strength condition is applied to establish a Volterra type integral equation, which characterizes the damage process for hereditary materials. These resulting differential equations are then solved with appropriate initial conditions. The study utilizes the obtained results to create time-dependent graphs of the damage parameter and performs comparisons. A piezoelectric defect detection device for polymer composite materials has been proposed through the application of mathematical modeling.

On pseudofrobenius imprimitive association schemes

An (association) scheme is said to be Frobenius if it is the (orbital) scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group G with abelian kernel to be determined up to isomorphism only by the character table of G. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with n vertices and Frobenius automorphism group is equal to 2 unless $$n\in \{p,p^2,p^3,pq,p^2q\}$$ , where p and q are distinct primes.

Characterizations of Some Groups in Terms of Centralizers

A group G is said to be n-centralizer if its number of element centralizers \(\mid {{\,\mathrm{Cent}\,}}(G)\mid =n\), an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element centralizers are abelian. For any non-abelian n-centralizer group G, we prove that \(\mid \frac{G}{Z(G)}\mid \le (n-2)^2\), if \(n \le 12\) and \(\mid \frac{G}{Z(G)}\mid \le 2(n-4)^{{log}_2^{(n-4)}}\) otherwise, which improves an earlier result. We prove that if G is an arbitrary non-abelian n-centralizer F-group, then gcd\((n-2, \mid \frac{G}{Z(G)}\mid ) \ne 1\). For a finite F-group G, we show that \(\mid {{\,\mathrm{Cent}\,}}(G)\mid \ge \frac{\mid G \mid }{2}\) iff \(G \cong A_4 \), an extraspecial 2-group or a Frobenius group with abelian kernel and complement of order 2. Among other results, for a finite group G with non-trivial center, it is proved that \(\mid {{\,\mathrm{Cent}\,}}(G)\mid = \frac{\mid G \mid }{2}\) iff G is an extraspecial 2-group. We give a family of F-groups which are not CA-groups and extend an earlier result.

A survey of Schreier-type extensions of monoids

We give an overview of a number of Schreier-type extensions of monoids and discuss the relation between them. We begin by discussing the characterisations of split extensions of groups, extensions of groups with abelian kernel and finally non-abelian group extensions. We see how these characterisations may be immediately lifted to Schreier split extensions, special Schreier extensions and Schreier extensions respectively. Finally, we look at weakenings of these Schreier extensions and provide a unified account of their characterisation in terms of relaxed actions.

Brauer–Manin obstruction to the local–global principle for the embedding problem

We study an analogue of the Brauer–Manin obstruction to the local–global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic) Brauer–Manin obstruction is the only one to strong approximation when the embedding problem has abelian kernel.

On Nonlinear Viscoelasticity of Elastomeric ISOLGOMMA Mats

This paper is devoted to an analysis of experimental results on elastomeric mats applied to the pavement on the underground station “Belgrade-junction” (“Beograd-ˇcvor” in Serbian). The elastomeric composite mat taken as a support of the underground tracks is called ISOLGOMMA-M40AV. For an appropriate modelling a nonlinear viscoelasticity model is needed. The Rabotnov’s nonlinear hereditary integral equation with Abelian kernel is applied and calibrated from triangular history stress controlled testing results.

On finite embedding problems with abelian kernels

Given a Hilbertian field k and a finite set S of Krull valuations of k, we show that every finite split embedding problem G→Gal(L/k) over k with abelian kernel has a solution Gal(F/k)→G such that every v∈S is totally split in F/L. Two applications are then given. Firstly, we solve a non-constant variant of the Beckmann–Black problem for solvable groups: given a field k and a non-trivial finite solvable group G, every Galois field extension F/k of group G is shown to occur as the specialization at some t0∈k of some Galois field extension E/k(T) of group G with E⊈k‾(T). Secondly, we contribute to inverse Galois theory over division rings, by showing that, for every division ring H and every automorphism σ of H of finite order, all finite semiabelian groups occur as Galois groups over the skew field of fractions H(T,σ) of the twisted polynomial ring H[T,σ].

Abelian kernels, profinite topologies and the extension problem

Abelian kernels, profinite topologies and the extension problem

Baer sums for a natural class of monoid extensions

It is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this characterization of extensions to a natural class of extensions of monoids, the cosetal extensions. An extension is cosetal if for all g,g' in G in which e(g) = e(g'), there exists a (not necessarily unique) n in N such that g = k(n)g'. These extensions generalise the notion of special Schreier extensions, which are themselves examples of Schreier extensions. Just as in the group case where a semidirect product could be associated to each extension with abelian kernel, we show that to each cosetal extension (with abelian group kernel), we can uniquely associate a weakly Schreier split extension. The characterization of weakly Schreier split extensions is combined with a suitable notion of a factor set to provide a cohomology group granting a full characterization of cosetal extensions, as well as supplying a Baer sum.

On the problems of controllability and uncontrollability for some mechanical systems described by the equations of vibrations of plates and beams with integral memory

Controllability problems for some models of plates and beams with integral memory are considered. The vibrational equation of the plate contains an Abelian kernel in the integral term, and the vibrational equation of the beam contains a continuous kernel consisting of a finite sum of decreasing exponential functions. It is proved that by controlling the whole domain, the first system cannot be driven to a state of rest, and for the second system, controllability to rest is possible.

Decompositions of locally compact contraction groups, series and extensions

A locally compact contraction group is a pair (G,α), where G is a locally compact group and α:G→G an automorphism such that αn(x)→e pointwise as n→∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,α) which are central extensions{0}→Fp((t))→G→Fp((t))→{0} of the additive group of the field of formal Laurent series over Fp=Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.

The Abelian Kernel of an Inverse Semigroup

The problem of computing the abelian kernel of a finite semigroup was first solved by Delgado describing an algorithm that decides whether a given element of a finite semigroup S belongs to the abelian kernel. Steinberg extended the result for any variety of abelian groups with decidable membership. In this paper, we used a completely different approach to complete these results by giving an exact description of the abelian kernel of an inverse semigroup. An abelian group that gives this abelian kernel was also constructed.

The Nine Lemma and the push forward construction for special Schreier extensions of monoids with operations

We show that the Nine Lemma holds for special Schreier extensions of monoids with operations. This fact is used to obtain a push forward construction for special Schreier extensions with abelian kernel. This construction permits to give a functorial description of the Baer sum of such extensions.

On the embedding of central extensions into permutation wreath products

We find a necessary condition for embedding a central extension of a group G with elementary abelian kernel into the wreath product that corresponds to a given permutation action of G.

On Some Results of a Torsion-Free Abelian Kernel Group

On Some Results of a Torsion-Free Abelian Kernel Group

The Grunwald problem and approximation properties for homogeneous spaces

Given a group $G$ and a number field $K$, the Grunwald problem asks whether given field extensions of completions of $K$ at finitely many places can be approximated by a single field extension of $K$ with Galois group G. This can be viewed as the case of constant groups $G$ in the more general problem of determining for which $K$-groups $G$ the variety $\mathrm{SL}_n/G$ has weak approximation. We show that away from an explicit set of bad places both problems have an affirmative answer for iterated semidirect products with abelian kernel. Furthermore, we give counterexamples to both assertions at bad places. These turn out to be the first examples of transcendental Brauer-Manin obstructions to weak approximation for homogeneous spaces.

On normal subgroups with consecutive G-class sizes

Let [Formula: see text] be a group and [Formula: see text] be a normal subgroup of [Formula: see text]. If the set [Formula: see text] is composed by consecutive integers, then [Formula: see text] is either nilpotent or a quasi-Frobenius group with abelian kernel and complements. This is a generalization of Theorem 2 of [A. Beltrán, M. J. Felipe and C. G. Shao, [Formula: see text]-divisibility of conjugacy class sizes and normal [Formula: see text]-complements, J. Group Theory 18 (2015) 133–141].

Baer sums of special Schreier extensions of monoids

We show that the special Schreier extensions of monoids, with abelian kernel, admit a Baer sum construction, which generalizes the classical one for group extensions with abelian kernel. In order to do that, we characterize the special Schreier extensions by means of factor sets.