Related papers: On asymorphisms of groups

We show that if $\kappa \leq \omega$ and there exists a group topology without non-trivial convergent sequences on an Abelian group $H$ such that $H^n$ is countably compact for each $n<\kappa$ then there exists a topological group $G$ such…

For every infinite cardinal $\kappa$ with $\kappa^+=2^\kappa$ we construct a group $G$ of cardinality $|G|=\kappa^+$ such that (i) $G$ is $36$-Shelah, which means that $A^{36}=G$ for any subset $A\subseteq G$ of cardinality $|A|=|G|$; (ii)…

Every infinite group $G$ of regular cardinality can be partitioned $G=A_1\cup A_2$ so that $G\neq FA_1$, $G\neq FA_2$ for every subset $F\subset G$ of cardinality $|F|<|G|$. The first author asked whether the same is true for each group $G$…

By definition, the sharp packing index $\ind_P^\sharp(A)$ of a subset $A$ of an abelian group $G$ is the smallest cardinal $\kappa$ such that for any subset $B\subset G$ of size $|B|\ge\kappa$ the family $\{b+A:b\in B\}$ is not disjoint. We…

In 1990, Comfort asked: is there, for every cardinal number $\alpha \leq 2^{\mathfrak{c}}$, a topological group $G$ such that $G^\gamma$ is countably compact for all cardinals $\gamma<\alpha$, but $G^\alpha$ is not countably compact? A…

Let $G$ and $H$ be locally compact groups and consider their associate spaces of almost periodic functions $AP(G)$ and $AP(H)$. We investigate the continuous group homomorphisms induced by isometries of $AP(G)$ into $AP(H)$. Among others,…

In S. 1 we deal with amalgamation bases, e.g., we define when an a.e.c. $k$ has $(\lambda,\kappa)$-amalgamation which means "many" M in $K^k_\lambda$ are amalgamation bases. We then consider what happens for the class of lf groups. In S. 2…

Let \lambda be a cardinal with \lambda=\lambda^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality \lambda and characteristic p such that the automorphism group of K_0 is a free group of…

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property…

The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M…

For certain uncountable cardinals $\kappa$ we produce a group of cardinality $\kappa$ which is freely indecomposable, strongly $\kappa$-free, and whose abelianization is free abelian of rank $\kappa$. The construction takes place in…

We equip the product of countably many copies of a compact Abelian group X with the uniform topology, and study some properties of the topological group G thus obtained. In particular, we determine the cardinality of the dual group of G,…

On every set A there is a rigid binary relation i.e. such a relation R \subseteq A \times A that there is no homomorphism (A,R) \rightarrow (A,R) except the identity (Vop{\v{e}}nka et al. [1965]). We prove that for each infinite cardinal…

Given any $\lambda\leq\kappa$, we construct a symmetric extension in which there is a set $X$ such that $\aleph(X)=\lambda$ and $\aleph^*(X)=\kappa$. Consequently, we show that $\mathsf{ZF}+$"For all pairs of infinite cardinals…

(1) Every infinite, Abelian compact (Hausdorff) group K admits 2^|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact. (2) Every infinite Abelian group G admits a…

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$…

Let B(kappa, lambda) be the subalgebra of P(kappa) generated by [kappa]^{<= lambda}. It is shown that if B is any homomorphic image of B(kappa, lambda) then either |B|< 2^lambda or |B|=|B|^lambda, moreover if X is the Stone space of B then…

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

A subset $R$ of the vertex set of a graph $\Gamma$ is said to be $(\kappa,\tau)$-regular if $R$ induces a $\kappa$-regular subgraph and every vertex outside $R$ is adjacent to exactly $\tau$ vertices in $R$. In particular, if $R$ is a…

Let $A \leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = \{1\}$. The \emph{classifying complements problem} asks for the description and classification of all…