We study generic properties of topological groups in the sense of Baire category.

First, we investigate countably infinite groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed groups and the word problem. Recently, I. Goldbring, S. Kunnawalkam Elayavalli, and Y. Lodha proved that every isomorphism class is meager among countably infinite groups. In contrast, it follows from the work of W. Hodges on model-theoretic forcing that there exists a comeager isomorphism class among countably infinite abelian groups. We present a new elementary proof of this result.

Then, we turn to compact metrizable abelian groups. We use Pontryagin duality to show that there is a comeager isomorphism class among compact metrizable abelian groups. We discuss its connections to the countably infinite case.

Finally, we study compact metrizable groups. We prove that the generic compact metrizable group is neither connected nor totally disconnected; also it is neither torsion-free nor a torsion group.

Let $M(A,n)$ be the Moore space of type $(A,n)$ for an Abelian group A and $n\ge 2$ . We show that the loop space $\Omega (M(A,n))$ is homotopy nilpotent if and only if A is a subgroup of the additive group $\mathbb {Q}$ of the field of rationals. Homotopy nilpotency of loop spaces $\Omega (M(A,1))$ is discussed as well.

We give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on Burnside’s lemma applied to the action of the power automorphism group. The resulting formula generalises Menon’s identity.

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik–Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A / pA is infinite and for every prime p, there are only finitely many natural numbers n such that $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.

Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.

For any finite abelian group $G$ with $|G|=m$, $A\subseteq G$ and $g\in G$, let $R_{A}(g)$ be the number of solutions of the equation $g=a+b$, $a,b\in A$. Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016, arXiv:1612.08722v1] proved that, if $m\geq 36$ and $R_{A}(n)\geq 1$ for all $n\in \mathbb{Z}_{m}$, then there exists $n\in \mathbb{Z}_{m}$ such that $R_{A}(n)\geq 6$. In this paper, for any finite abelian group $G$ with $|G|=m$ and $A\subseteq G$, we prove that (a) if the number of $g\in G$ with $R_{A}(g)=0$ does not exceed $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$, then there exists $g\in G$ such that $R_{A}(g)\geq 6$; (b) if $1\leq R_{A}(g)\leq 6$ for all $g\in G$, then the number of $g\in G$ with $R_{A}(g)=6$ is more than $\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$.

We answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., $\operatorname{cov}\left( \text{M} \right)=\mathfrak{c})$, a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

We give a model-theoretic proof of the fact that for all infinite Abelian groups P ≠ NP in the sense of binary nondeterminism. This result has been announced 1994 by Christine Gaßner.

A group A is said to be endoprimal if its term functions are precisely the functions which permute with all endomorphisms of A. In this paper we describe endoprimal groups in the following three classes of abelian groups: torsion groups, torsionfree groups of rank at most 2, direct sums of a torsion group and a torsionfree group of rank 1.

We show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

The article studies the class of abelian groups G such that in every direct sum decomposition G = A ⊕ B, A is 5-projective. Such groups are called pds groups and they properly include the quasi-projective groups.

The pds torsion groups are fully determined.

The torsion-free case depends on a lemma that establishes freedom in the non-indecomposable case for several classes of groups. There is evidence suggesting freedom in the general reduced torsion-free case but this is not established and prompts a logical discussion. It is shown, for example, that pds torsion-free groups must be Whitehead if they are not indecomposable, but that there exists Whitehead groups that are not pds if there exist non-free Whitehead groups.

The mixed case is characterized and examples are given.

An ideal I defined on a group G is called idempotent if for every A ∈ I, {g ∈ G:Ag−1 ∉ ∈ I} ∈ I. We show that a countably complete idempotent ideal on an abelian group cannot be prime but may have strong saturation properties.