On the structure of abelian profinite groups

Abstract

A subgroup $G$ of a product $\prod _{i\in \mathbb{N}}{G}_i$ is \emph{rectangular} if there are subgroups $H_i$ of $G_i$ such that $G=\prod _{i\in \mathbb{N}}{H}_i$. We say that $G$ is \emph{weakly rectangular} if there are finite subsets $F_i\subseteq \mathbb{N}$ and subgroups $H_i$ of $\underset{j\in F_i}{⨁}{G}_j$ that satisfy $G=\prod _{i\in \mathbb{N}}{H}_i$. %We say that $G$ is a \emph{subdirect product} of the family $\{G_i{\}}_{i\in I}$ if $G$ is weakly rectangular and %$G\cap \underset{i\in I}{⨁}{G}_i=\underset{i\in \mathbb{N}}{⨁}{H}_i$. In this paper we discuss when a closed subgroup of a product is weakly rectangular. Some possible applications to the theory of group codes are also highlighted.