Pure Subgroups Research Articles

Pure infinitely braided Thompson groups

We show that pure subgroups of infinitely braided Thompson’s are bi-orderable. For every finitely generated pure subgroup, we give explicit sets of generators.

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

The virtual braid group VBn, the virtual twin group VTn and the virtual triplet group VLn are extensions of the symmetric group Sn, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group Sn are the pure virtual braid group VPn, the pure virtual twin group PVTn and the pure virtual triplet group PVLn, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group PVLn, the commutator subgroup VTn′ of VTn and the commutator subgroup VLn′ of VLn. Our results complete the understanding of these groups, except that of VBn′, for which the existence of a finite presentation is not known for n≥4. We also prove that VLn/PVLn′ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.

Neat locally compact abelian p-groups

AbstractInspired by a result of Kulikov about splitting of pure subgroups in discrete abelian p-groups we classify all locally compact abelian p-groups with no elements of infinite p-height in which every maximal monothetic subgroup splits. Descriptions of separable compact groups with elements of finite p-height, in which every pure monothetic subgroup splits, are obtained.

Isomorphisms and commensurability of surface Houghton groups

Abstract We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.

Congruence subgroups and crystallographic quotients of small Coxeter groups

Abstract Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n {S_{n}} has three natural extensions, namely the braid group B n {B_{n}} , the twin group T n {T_{n}} and the triplet group L n {L_{n}} . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n {T_{n}} and L n {L_{n}} when n ≥ 4 {n\geq 4} . We also determine subquotients of principal congruence subgroups of T n {T_{n}} , and identify the pure twin group P ⁢ T n {PT_{n}} and the pure triplet group P ⁢ L n {PL_{n}} with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ {T_{n}/PT_{n}^{\prime}} , T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} and L n / P ⁢ L n ′ {L_{n}/PL_{n}^{\prime}} are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} .

On pure subgroups and direct summands of abelian groups

On pure subgroups and direct summands of abelian groups

Pure subgroups of non-abelian groups

Pure subgroups of non-abelian groups

Property 𝑅∞ for some spherical and affine Artin–Tits groups

Abstract Let n ⩾ 2 n\geqslant 2 . In this note, we give a short uniform proof of property R ∞ R_{\infty} for the Artin–Tits groups of spherical types A n A_{n} , B n B_{n} , D 4 D_{4} , I 2 ⁢ ( m ) I_{2}(m) ( m ⩾ 3 m\geqslant 3 ), their pure subgroups, and for the Artin–Tits groups of affine types A ~ n - 1 \tilde{A}_{n-1} and C ~ n \tilde{C}_{n} . In particular, we provide an alternative proof of a recent result of Dekimpe, Gonçalves and Ocampo, who established property R ∞ R_{\infty} for pure Artin braid groups.

A lattice-theoretic characterization of pure subgroups of Abelian groups

Let G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice mathcal {L}(H) embeds in mathcal {L}(G). It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

On The Intersection of Semi-Pure Subgroups of Abelian Group

The following question was raised by L.Fuchs: "what are the subgroups of an abelian group G that can be represented as intersections of pure subgroups of G ? . Fuchs also added that “One of my main aims is to give the answers to the above question". In this paper, we shall define new subgroups which are a family of the pure subgroups. Then we shall answer problem 2 of L.Fuchs by these semi-pure subgroups which can be represented as the intersections of pure subgroups.

On a problem of Călugăreanu, Chekhlov and Krylov regarding pure endomorphic images of abelian groups

Abstract A problem of Călugăreanu, Chekhlov and Krylov asks which abelian groups have the property that all of their pure subgroups that are endomorphic images are necessarily summands. Complete answers are given for the groups in certain classes (for example, the torsion groups). On the other hand, examples are constructed that show a complete solution to the general problem is likely to be quite difficult.

On The Almost almost Dense and Pure-4 Subgroups of Abelian Groups

We gave the some general properties of anew subgroups which are called pure (1-2-3) subgroups of abelian group G. And he gave some new results of minimal neat subgroups of abelian groups. “L. Fuchs” poses the problem of characterizing the subgroups of an abealian group G which are intersections of finitely many pure subgroups of G (problem 13, p.134).This problem has been solved by “Khalid ben abdullah” and John Irwin.In this paper we shall give the generalization of the problem solved by Khalid Ben abdullah. Firstly we shall give the definition of new subgroups which are a some family of the pure subgroups, and we shall study the some general properties of this new type of the purity.

Algebraic description of limit models in classes of abelian groups

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: Theorem 0.1(1)If G is a limit model of cardinality λ in the class of abelian groups with the subgroup relation, thenG≅(⊕λQ)⊕⊕pprime(⊕λZ(p∞)).(2)If G is a limit model of cardinality λ in the class of torsion-free abelian groups with the pure subgroup relation, then:•If the length of the chain has uncountable cofinality, thenG≅(⊕λQ)⊕Πpprime(⊕λZ(p))‾.•If the length of the chain has countable cofinality, then G is not algebraically compact. We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and (<ℵ0)-tame and short.

A Novel Combination of Prion Strain Co-Occurrence in Patients with Sporadic Creutzfeldt-Jakob Disease

Six subgroups of sporadic Creutzfeldt-Jakob disease have been identified by distinctive clinicopathologic features, genotype at polymorphic codon 129 [methionine (M)/valine (V)] of the PRNP gene, and type of abnormal prion proteins (type 1 or 2). In addition to the pure subgroups, mixed neuropathologic features and the coexistence of two types of abnormal prion proteins in the same patient also have beenreported. Here, we found that a portion of the patients previously diagnosed as MM1 had neuropathologic characteristics of the MM2 thalamic form (ie, neuronal loss of the inferior olivary nucleus of the medulla). Furthermore, coexistence of biochemical features of the MM2 thalamic form also was confirmed in the identified cases. In addition, in transmission experiments using prion protein-humanized mice, the brain material from the identified case showed weak infectivity and generated characteristic abnormal prion proteins in the inoculated mice resembling those after inoculation with brain material of MM2 thalamic form. Taken together, these results show that the co-occurrence of MM1 and MM2 thalamic form is a novel entity of sporadic Creutzfeldt-Jakob disease prion strain co-occurrence. The present study raises the possibility that the co-occurrence of the MM2 thalamic form might have been overlooked so far because of the scarcity of abnormal prion protein accumulation and restricted neuropathology.

Locally compact abelian p-groups

A locally compact abelian group is called periodic if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as–decomposing them into local products of their Sylow p- subgroups,–providing new descriptions of periodic abelian torsion groups, and of–periodic abelian divisible groups and their torsion-free and their torsion components,–reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent p-groups,–providing a definition of a general p-rank for all locally compact abelian p-groups.

Subgroups generated by images of endomorphisms of Abelian groups and duality

Abstract A subgroup H of a group G is called endo-generated if it is generated by endo-images, i.e. images of endomorphisms of G. In this paper we determine the following classes of Abelian groups: (a) the endo-groups, i.e. the groups all of whose subgroups are endo-generated; (b) the endo-image simple groups, i.e. the groups such that no proper subgroup is an endo-image; (c) the pure-image simple, i.e. the groups such that no proper pure subgroup is an endo-image; (d) the groups all of whose endo-images are pure subgroups; (e) the ker-gen groups, i.e. the groups all of whose kernels are endo-generated. Some dual notions are also determined.

On loxodromic actions of Artin–Tits groups

Artin–Tits groups act on a certain delta-hyperbolic complex, called the “additional length complex”. For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mapping class groups, we conjecture that “most” elements of Artin–Tits groups act loxodromically. More precisely, in the Cayley graph of a subgroup G of an Artin–Tits group, the proportion of loxodromically acting elements in a ball of large radius should tend to one as the radius tends to infinity. In this paper, we give a condition guaranteeing that this proportion stays away from zero. This condition is satisfied e.g. for Artin–Tits groups of spherical type, their pure subgroups and some of their commutator subgroups.

ON DIRECT PRODUCT pure − 1 − 2 − 3 SUBGROUPS IN ABELIAN GROUP Gn×Gm

In this paper, we shall define new subgroups which are called pure− 1 − 2 − 3 in abelian groups Gn×Gm for all n,m ∈ N which are a family of pure subgroups. In[1],[2]H.M.A.Abdullah gave the some general properties of pure− 1 − 2 − 3 in abelian group G , but here, we shall prove more than properties on this subgroups in Mod Gn×Gm , which are not valid for pure subgroups.

Abelian groups whose nonzero endomorphisms have nonessential kernels

In this paper, we describe completely the [Formula: see text]-singular subgroup of an abelian group and a [Formula: see text]-nonsingular abelian group in terms of the basic subgroups of its [Formula: see text]-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a [Formula: see text]-nonsingular abelian group are [Formula: see text]-nonsingular and give a condition under which a pure extension of a [Formula: see text]-nonsingular abelian group by a [Formula: see text]-nonsingular group is [Formula: see text]-nonsingular.

Association between GABA3 Gene Polymorphisms and Attention Deficit Hyperactivity Disorder in Korean Children.

ObjectiveAttention deficit hyperactivity disorder (ADHD) is common disorder of the school-age population. ADHD is familial and genetic studies estimate heritability at 80–90%. The aim of the present study was to investigate the association between the genetic type and alleles for gamma-aminobutyric acid receptor subunit beta-3 (GABA3) gene in Korean children with ADHD.MethodsThe sample consisted of 180 ADHD children and 159 control children. We diagnosed ADHD according to DSM-IV. ADHD symptoms were evaluated with Conners' Parent Rating Scales and Dupaul Parent ADHD Rating Scales. Blood samples were taken from the 339 subjects, DNA was extracted from blood lymphocytes, and PCR was performed for GABA3 rs2081648, rs1426217 and rs981778 Polymorphism. Alleles and genotype frequencies were compared using the chi-square test. We compared the allele and genotype frequencies of GABA3 gene polymorphism in the ADHD and control groups.ResultsThis study showed that there was a significant correlation among the frequencies of the rs2081648 (OR=0.71, 95% CI=0.51–0.98, p=0.040) of alleles of MAO, but the final conclusions are not definite. Follow up studies with larger patient or pure subgroups are expected.ConclusionThese results suggested that GABA3 might be related to ADHD symptoms.