In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T=\mathbf{R}/\mathbf{Z}$ by subsets of $\mathbf{Z}$. Here we consider new types of subgroups: let $K\subseteq T$ be a Kronecker set (a compact set on which every continuous function $f:K\to T$ can be uniformly approximated by characters of $T$), and $G$ the group generated by $K$. We prove (Theorem 1) that $G$ can be characterized by a subset of ${\mathbf{Z}}^2$ (instead of a subset of $\mathbf{Z}$). If $K$ is finite, Theorem 1 implies our...

In this paper we have given the construction of free $\ell$-groups generated by a po-group and the construction of free products in any sub-product class $𝒰$ of $i\ell$-groups. We have proved that the $𝒰$-free products satisfy the weak subalgebra property.

Let $G\cong C_{n_1}\oplus ...\oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1·...·w_n$ is a sequence of integers, all but at most one relatively prime to $\left|G\right|$, and $S$ is a sequence over $G$ with $\left|S\right|\ge \left|W\right|+\left|G\right|-1\ge \left|G\right|+1$, the maximum multiplicity of $S$ at most $\left|W\right|$, and $\sigma \left(W\right)\equiv 0mod\left|G\right|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\underset{i=1}{\sum ^n}{w}_i{s}_i$, with $s_1·...·s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...