Subdirect Products of Finite Abelian Groups

Abstract

A subgroup G of a product $\prod \limits _{i\in \mathbb {N}}G_i$ is rectangular if there are subgroups $H_i$ of $G_i$ such that $G=\prod \limits _{i\in \mathbb {N}}H_i$. We say that G is weakly rectangular if there are finite subsets $F_i\subseteq \mathbb {N}$ and subgroups $H_i$ of $\bigoplus \limits _{j\in F_i} G_j$ that satisfy $G=\prod \limits _{i\in \mathbb {N}}H_i$. 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.