Some finite abelian group theory and some q-series identities

TL;DR

This paper introduces new identities related to finite abelian groups and q-series, expanding on classical results by Hall and Cohen-Lenstra, with potential applications in algebra and number theory.

Contribution

It generalizes existing identities for finite abelian groups and presents a novel q-series identity, broadening the theoretical framework in algebraic combinatorics.

Findings

New identities for subposets of finite abelian groups

A novel q-series identity derived in the paper

Extensions of classical Hall and Cohen-Lenstra identities

Abstract

For a fixed odd prime $\ell$, we present new families of identities defined on various subposets of the poset of isomorphism classes of finite abelian $\ell$-groups, generalizing identities of Hall and Cohen-Lenstra. We also present a $q$-series identity.