Related papers: Minimal pseudocompact group topologies on free abe…

We obtain necessary and sufficient conditions when a pseudocompact paratopological group is topological. (2-)pseudocompact and countably compact paratopological groups that are not topological are constructed. It is proved that each…

A topological group $X$ is defined to have $compact$ $exponent$ if for some number $n\in\mathbb N$ the set $\{x^n:x\in X\}$ has compact closure in $X$. Any such number $n$ will be called a compact exponent of $X$. Our principal result…

Let G be an abelian group and let lambda be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If lambda is uncountable, then G has lambda pairwise disjoint, non-free subgroups. There is an example where…

Let $G$ be a finite group, and let $\kappa(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\kappa(G) \geq…

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is…

Let $G$ be a finite non-abelian group and $\kappa_1(G)$ the number of conjugate classes of minimal non-abelian subgroups of $G$. The structure of $G$ with $\kappa_1(G)=1$ is determined. In the case of $G$ being the $p$-groups, the structure…

Given a set $\mathcal{F}$ of finite groups, it is said that a group $G$ is an $\mathcal{F}$-cover if every group in $\mathcal{F}$ is isomorphic to a subgroup of $G$. Moreover, $G$ is a minimum $\mathcal{F}$-cover if there is no…

We prove an "abelian, locally compact" Whitehead theorem in fine shape: A fine shape morphism between locally connected finite-dimensional locally compact separable metrizable spaces with trivial $\pi_0$ and $\pi_1$ is a fine shape…

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is…

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 $\kappa$…

Assuming the existence of $\mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $\mathfrak c$ that admit a countably compact group topology. We show that for each $\kappa \in [\mathfrak…

Let $(X,\omega)$ be a compact symplectic manifold of dimension $2n$ and let $Ham(X,\omega)$ be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant $C$, depending on $X$ but not on $\omega$, such that any finite…

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy…

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…

We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$…

A.V.Arkhangel'skii asked in 1981 if the variety $\mathfrak V$ of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued…

In this paper we develop the homological version of $\Sigma$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as…

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above…

A topological group $(G,\mu)$ from a class $\mathcal G$ of MAP topological abelian groups will be called a {\it Mackey group} in $\mathcal G$ if it has the following property: if $\nu$ is a group topology in $G$ such that $(G,\nu)\in…

The Gromov-Lawson-Rosenberg conjecture for a group G states that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. It is known to be true…