Uncountable Abelian Group Research Articles

Transformations of the transfinite plane

Abstract We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $\kappa $ , if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $ , there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$ , there exist $x\neq y$ in X such that $f(x+y)=g$ .

From uncountable abelian groups to uncountable nonabelian groups

The paper is a survey of my research related to applications of abelian groups to non-commutative groups some of which are direct ones, but the emphasis lies on indirect ones.

Every Uncountable Abelian Group Admits a Nonnormal Group Topology

Every Uncountable Abelian Group Admits a Nonnormal Group Topology

Every uncountable abelian group admits a nonnormal group topology

If G is a locally compact Abelian group, let G + {{\mathbf {G}}^ + } denote the underlying group of G equipped with the weakest topology that makes all the continuous characters of G continuous. Thus defined, G + {{\mathbf {G}}^ + } is a totally bounded topological group. We prove: Theorem. G + {{\mathbf {G}}^ + } is normal if and only if G is σ \sigma -compact. When G is discrete, this theorem answers in the negative a question posed in 1990 by E. van Douwen, and it partially solves a problem posed in 1945 by A. Markov.

Counting subgroups and topological group topologies

Algebra. Either |^(G)| = 2a or |^(G)| = a. If \S?(G) = a then a = co. We describe and characterize those (countable) G such that \S?(G) = ω, and we give several examples. Topology. If γ 2 α, then &(y) = 0; otherwise 2 αγ . If γ > 2 α then Jt(y) = 0; if log(α) < γ < 2 α then = 2 α γ; and if ω < γ < a then K(γ)| = 2 α. 0. Introduction and motivation. As a reading of the Synopsis may suggest, this work originated with the authors' interest in the following questions: Given an infinite Abelian group, how many topological group topologies does G possess? Of these, how many may be chosen pairwise non-homeomorphic? How many metrizable? How many totally bounded? How many totally bounded and metrizable? We approached the latter questions through a result from [6] which gives a one-to-one order-preserving correspondence between the set SS(G) of totally bounded topological group topologies for G and the set of point-separat ing subgroups of the homomorphism group Hom(G, T). Thus it became natural—indeed necessary—to count the number of subgroups of a group of the form Hom(G,T). In §§1 and 2, which we believe have algebraic interest quite independent of their topological roots, we do a bit more: We show that every uncountable Abelian group G has 2|G| subgroups, and we describe in some detail the fine algebraic structure of what we call ω-groups. These are by definition the (necessarily countable) Abelian groups G with fewer then 2|G| subgroups; we show that each ω-group has exactly ω-many subgroups, and we describe the relationship between the ω-groups and the so-called q.d. groups of Beaumont and Pierce [2]. The algebraic analysis of §1, together with the result cited from [6], allows us to describe some gross features of the partially ordered sets £%(G). Here our work is sufficiently coarse that the various cardinal

Whitehead groups may not be free even assuming ch, II

We prove some theorems on uncountable abelian groups, and consistency results promised in the first part, and also that a variant of $$ \diamondsuit _{\omega _1 } $$ called ♣ (club), is consistent with 2ℵ 0<ℵ1.

On uncountable abelian groups

We continue the investigation from [10], [11], [12] on uncountable abelian groups. This paper tends more to group theory and was motivated by Nunke’s statement (in [9]) that Whitehead problem, rephrased properly, is not solved yet.