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.

For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of $\operatorname {stp}(ac)$ is abelian, and the underlying theory is G-compact, then the Lascar groups of $\operatorname {stp}(ac)$ and of $\operatorname {stp}(a)$ are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of $\operatorname {stp}(ac)$ is abelian, nor the assumption of c being finite can be removed.

The aim of this paper is to present some results about the space $L^{\varPhi }(\nu ),$ where $\nu$ is a vector measure on a compact (not necessarily abelian) group and $\varPhi$ is a Young function. We show that under natural conditions, the space $L^{\varPhi }(\nu )$ becomes an $L^{1}(G)$-module with respect to the usual convolution of functions. We also define one more convolution structure on $L^{\varPhi }(\nu ).$

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and let $H$ be a closed subgroup of $G$. Let ${G}/{H}\;$ be the left coset space of $H$ in $G$ and let $\mu$ be the normalized $G$-invariant measure on ${G}/{H}\;$ associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space ${{L}^{2}}\left( {G}/{H,\,\mu }\; \right)$.

In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

In this paper, the amenability and approximate amenability of weighted ℓp-direct sums of Banach algebras with unit, where 1≤p<∞, are completely characterized. Applications to compact groups and hypergroups are given.

The main purpose of this paper is to study the validity of the Hausdorff–Young inequality for vector-valued functions defined on a non-commutative compact group. As we explain in the introduction, the natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some important questions are posed.