Finite Abelian Groups with Toroidal Subgroup Lattices

TL;DR

This paper investigates the subgroup lattice structures of finite abelian groups, determining their genus and classifying those that can be embedded into a torus, thus linking algebraic and topological properties.

Contribution

It provides a classification of finite abelian groups with subgroup lattices embeddable into the torus and calculates their genus, advancing understanding of their topological lattice properties.

Findings

Identified the genus of subgroup lattices for specific abelian groups

Classified all finite abelian groups with subgroup lattices embeddable into the torus

Established connections between algebraic group properties and topological embeddings

Abstract

In this paper, we determine the genus of the subgroup lattice of several families of abelian groups. In doing so, we classify all finite abelian groups whose subgroup lattices can be embedded into the torus.