Some Finite Abelian Group Theory and Some $${}$$-Series Identities

The purpose of this article is to compute certain weighted sums over various subposets of the poset of isomorphism classes of finite abelian ({ell})-groups, where ({ell}) is a fixed odd prime. These sums are similar to previous sums computed by Hall and Cohen-Lenstra. The computation expands upon the previous analysis of Cohen-Lenstra while also using the tools of hypergeometric functions and ({q})-series. The identities computed in this article, while aesthetically pleasing in their own right, also turn out to be applicable to the construction of a random matrix version of the Cohen-Lenstra-Martinet heuristics.