I recently came across a construction that, in abstraction, leads to the following family of abelian groups: Fix $1<q<p$ with $q$ and $p$ relatively prime. The group $G_{(p,q)}$ is given by the presentation $$<g_0, g_1, g_2, \dots \mid g_i^p=g_{i+1}^q,\ g_ig_j=g_jg_i>.$$ In retrospect, I quickly realized that none of my training in group theory covered infinitely-generated abelian groups. So for now I've picked up Fuch's "Infinite Abelian Groups" but I was hoping that perhaps $G_{(p,q)}$ was already a known group with some literature specific to it. Does anyone recognize this group?

If it helps, one of the main properties of the groups is that $G_{(p,q)}/<\!g_k\!> \cong \mathbb{Z}/p^k\mathbb{Z}$. (Edit: This holds because of the relatively prime condition.)

EDIT: Other properties to consider:

$G_{(p,1)} \cong \mathbb{Z}$ under the map $g_i \mapsto p^i$ and $G_{(p,0)} \cong \bigoplus_{i=0}^\infty \mathbb{Z}/p\mathbb{Z}$. Hence the restriction $1<q$.