Bounded Rank Subgroups of Coxeter Groups, Artin Groups and One-Relator Groups with Torsion

We obtain a number of results regarding the freeness of subgroupsof Coxeter groups, Artin groups and one-relator groups withtorsion. In the case of Coxeter groups, we also obtain resultson quasiconvexity and subgroup separability. 2000 MathematicsSubject Classification 20F65, 20F55, 20F36, 20F06.     

Conjugacy and Other Properties of One-Relator Groups

In this note, we prove that certain one-relator groups are residually finite and have solvable (power) conjugacy problem, by an examination of the co-primeness of the exponent sum of some of the generators appearing in the relator.     

交换群上Hopf路余代数的结构分类

设G是群, kG是域k上的群代数. 对任意Hopf箭向Q=(G, r), 利用右kZ_u(C) -模的直积范畴∏_C∈K(G) MkZ_u(C)与kG-Hopf双模范畴^kG_kG M^kG_kG之间的同构, 可由^u(C)(kQ₁)¹上的右kZ_u(C) -模结构导出在箭向余模kQ₁上的kG-Hopf双模结构. 该文讨论在群G分别是2阶循环群与克莱茵四元群时的Hopf路余代数kQ^c的同构分类及其子Hopf代数kG[kQ₁]结构.     

A Freiheitssatz for Whitehead Graphs of One-Relator Groups with Small Cancellation

In this work we study one-relator groups with a certain small cancellation condition. We focus on the following two general problems: the free subgroups of these groups, and what can be said on the automorphism group of these groups. Both problems are widely open. We introduce a graph-theoretical test which, if successful, implies that the subgroup under consideration is free. We also extend a result due to V. Shpilrain on automorphism groups of one-relator groups.     

Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

In this article we prove comparative results on the Gromov hyperbolicity of plane domains equipped with the quasihyperbolic metric. By a comparative result we mean one which assumes hyperbolicity in one domain and obtains it in a different domain somehow related to the original one. We derive a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a plane domain Ω* obtained by deleting from the original domain Ω any uniformly separated union of compact sets. We present as well a result about stability of hyperbolicity.     

Abel群的亚同态的刻画

In this paper,a characterization of metahomomorphisrns on Abeliangroups is given.     

A Class of Strongly Decomposable Abelian Groups

Let G be a completely decomposable torsion-free Abelian group and G= G_i, where G _i is a rank 1 group. If there exists a strongly constructive numbering of G such that (G,) has a recursively enumerable sequence of elements g _i G _i , then G is called a strongly decomposable group. Let pi, i, be some sequence of primes whose denominators are degrees of a number p _i and let . A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that for some numbers i ₁,...,i _k . We bring in the concept of a quasihyperhyperimmune set, and specify a necessary and sufficient condition on the characteristic of A subject to which the group in question is strongly decomposable. Also, it is proved that every hyperhyperimmune set is quasihyperhyperimmune, the converse being not true.     

A Classification of Commutative Reduced Filial Rings

A ring R is said to be filial when for every I, J, if I is an ideal of J and J is an ideal of R then I is an ideal of R. The classification of commutative reduced filial rings is given.     

Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (A_n) is considered, where A₀ = A and A_n+1 = End A_n. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A₀ or after A₁.

     

Abelian Groups,Gauss Periods,and Normal Bases

A result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity.     

A Note on Finitely A-Presented Abelian Groups

The class of mixed groups  was introduced by Glaz and Wickless. This article investigates subclasses  of  such that the functor Hom(A, ?)/t Hom(A, ?) between  and the category of right E(A)/tE(A)-modules is full.     

Abelian Groups and Regular Modules

The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.     

Ultrafilters and Decompositions of Abelian Groups

We prove that every PS-ultrafilter on a group without second-order elements is a Ramsey ultrafilter. For an arbitrary PS-ultrafilter on a countable group G, we construct a mapping f: G such that f() is a P-point in the space *. We determine a new class of subselective ultrafilters, which is considerably wider than the class of PS-ultrafilters.     

Boolean Algebras,Generalized Abelian Rings,and Grothendieck Groups

ABSTRACT

A ring R is called generalized Abelian if for each idempotent e in R, eR and (1 ? e)R have no isomorphic nonzero summands. The class of generalized Abelian rings properly contains the class of Abelian rings. We denote by GAERS ? 1 the class of generalized Abelian exchange rings with stable range 1. In this article we prove, by introducing Boolean algebras, that for any R ∈ GAERS ? 1, the Grothendieck group K ₀(R) is always an Archimedean lattice-ordered group, and hence is torsion free and unperforated, which generalizes the corresponding results of Abelian exchange rings. Our main technical tool is the use of the ordered structure of K ₀(R)⁺, which provides a new method in the study of Grothendieck groups.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.