Normality in Free Topological Groups

Mathematical Notes - It is proved that if, for a subset A of a finite Abelian group G, under the action of a linear operator L: G3 → G2, the image L(A, A, A) has cardinality less than...     

Group Laws and Free Subgroups in Topological Groups

A proof is given that a permutation group in which differentfinite sets have different stabilizers cannot satisfy any grouplaw. For locally compact topological groups with this property,almost all finite subsets of the group are shown to generatefree subgroups. Consequences of these theorems are derived for:Thompson's group F, weakly branch groups, automorphism groupsof regular trees, and profinite groups with alternating compositionfactors of unbounded degree. 2000 Mathematics Subject Classification20B07, 20E10, 20E18, 20P05.     

关于几乎拓扑群的注记

In this paper, we mainly discuss some generalized metric properties and the cardinal invariants of almost topological groups. We give a characterization for an almost topological group to be a topological group and show that:(1) Each almost topological group that is of countable π-character is submetrizable;(2) Each left λ-narrow almost topological group isλ-narrow;(3) Each separable almost topological group is ω-narrow. Some questions are posed.     

Fuzzy Soft Topological Groups

Notions of Lowen type fuzzy soft topological space are introduced and some of their properties are established in the present paper. Besides this, a combined structure of a fuzzy soft topological space and a fuzzy soft group, which is termed here as fuzzy soft topological group is introduced. Homomorphic images and preimages are also examined. Finally, some definitions and results on fuzzy soft set are studied.     

Normed Topological Pseudovector Groups

A normed topological pseudovector group (NTPVG for short) is a valued topological group (V,?+?,||·||) (not necessarily Abelian) endowed with a continuous scalar multiplication \({\mathbb R}_+ \times V \ni (t,x) \mapsto t \cdot x \in V\) such that 0 ·x?=?e (e denotes the neutral element of V), 1 ·x?=?x, (st) ·x?=?s ·(t ·x), t ·(x?+?y)?=?(t ·x)?+?(t ·y) and ||t ·x||?=?t ||x|| for each t, \(s \in {\mathbb R}_+\) and x, y?∈?V. It is shown that every valued topological group can be isometrically and group-homomorphically embedded in a NTPVG as a closed subset by means of a functor. Locally compact NTPV groups are fully classified. It is shown that the (unbounded) Urysohn universal metric space can be endowed with a structure of a NTPV group of exponent 2.     

Multiplication Groups of Topological Loops

In this brief survey, we discuss the following question: Which Lie groups can occur as the multiplication groups Mult(L) of connected topological loops L? We describe correspondences between the structure of the group Mult(L) and the structure of the loop L.     

Free Internal Groups

In the first part of this note an elementary proof is given of the fact that algebraic functors, that is, functors induced by morphisms of Lawvere theories, have left adjoints provided that the category K\mathcal{K} in which the models of these theories take their values is locally presentable. The main focus however lies on the special cases of the underlying functor of the category Grp(K)\mathsf{Grp}(\mathcal{K}) of internal groups in K\mathcal{K} and the embedding of Grp(K)\mathsf{Grp}(\mathcal{K}) into Mon(K)\mathsf{Mon}(\mathcal{K}), the category of monoids in K\mathcal{K}: Here a unifying construction of the respective left adjoints is provided which not only works in case K\mathcal{K} is a locally presentable category but also when K\mathcal{K} is, for example, a particular category of topological spaces such as the category of Hausdorff or Tychonoff spaces or a cartesian closed topological category.     

On Subgroups and Quotient Groups of L-fuzzy Topological Groups

In paper [1], it gave the concept and basic properties of L-fuzzy topological groups. This paper will enter the concept of quotient groups of L-fuzzy topological groups and will prove some properties of subgroups and quotient groups. The persent work is the continuation of [1] and hence all the conventions in [1] still hold good in the present paper.     

The Small Index Property for Free Groups and Relatively Free Groups

1. Some groups with the small index property In this paper we shall use the methods of the paper [6] by Hodges,Hodkinson, Lascar and Shelah to show that the free group ofcountably infinite rank and certain relatively free groups ofcountably infinite rank have the small index property.     

Categorically Proper Homomorphisms of Topological Groups

We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the morphism level, study the stability properties of the newly defined types of maps, such as closure under direct products, and compare them with their counterparts in topology. We assume Hausdorffness only when our proofs require us to do so, which leads to new results and the affirmation of some facts that were known in a Hausdorff context.     

Weakly Linearly Compact Topological Abelian Groups

In this paper, we study linearly topological groups. We introduce the notion of a weakly linearly compact group, which generalizes the notion of a weakly separable group, and examine the main properties of such groups. For weakly linearly compact groups, we construct the character theory and present an algebraic characterization of some classes of such groups. Some well-known theorems for periodic Abelian groups are generalized for the case of linearly discrete, topological Abelian groups; for linearly compact and linearly discrete topological Abelian groups, we also construct the character theory and study some important properties of linearly discrete groups. For linearly discrete, topological Abelian groups, we analyze the splittability condition (Theorem 3.12) and present the characteristic condition of decomposability of a discrete group G into the direct sum of rank-1 groups. We also present an algebraic characterization of linearly compact groups. We introduce the notion of a weakly linearly compact, topological Abelian group, which generalizes the notion of a weakly separable Abelian group, and examine some properties of such groups. These groups are a generalization of fibrous Abelian groups introduced by Vilenkin. We give an algebraic characterization of divisible, weakly locally compact Abelian groups that do not contain nonzero elements of finite order (Proposition 7.9). For weakly locally compact Abelian groups, we construct universal groups.     

Homomorphisms from Automorphism Groups of Free Groups

The automorphism group of a finitely generated free group isthe normal closure of a single element of order 2. If m <n, then a homomorphism Aut(F_n)Aut(F_m) can have image of cardinalityat most 2. More generally, this is true of homomorphisms fromAut(F_n) to any group that does not contain an isomorphic imageof the symmetric group S_n+1. Strong restrictions are also obtainedon maps to groups that do not contain a copy of W_n = (Z/2)ⁿ S_n, or of Z^n–1. These results place constraints on howAut(F_n) can act. For example, if n 3, any action of Aut(F_n)on the circle (by homeomorphisms) factors through det : Aut(F_n)Z₂.2000 Mathematics Subject Classification 20F65, 20F28 (primary).     

Non-Hopfian Relatively Free Groups

To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960’s, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-Hopfian.     

Free Groups from Fields

In the nineteenth century, field theory brilliantly resolveda number of questions that had taxed mathematicians for centuries;for example, ‘The circle cannot be squared’ by straightedge and compass, and solving polynomial equations by radicalsis not always possible. These successes have continued to beheld up as superb examples of the power of mathematical thought,and are demonstrated at an undergraduate level. The purposeof this article is to provide another such natural example whichleads to a concrete realisation of the free group on 2 generators.     

拓扑群中广义度量性质的一个注记

主要讨论拓扑群中的一些广义度量性质.证明了对于拓扑群G和G的局部紧度量子群H,如果商群G/H是层空间(半层空间,κ半层空间,σ空间),则G也是层空间(半层空间, κ半层空间,σ空间),这肯定回答了Arhangel'skii A.V.和Uspenskij V.V.提出的一个问题.同时还讨论了弱拟第一可数的,不含Sω的闭拷贝的仿拓扑群.