Persistent homology of groups

TL;DR

This paper explores the application of persistent homology to p-groups, demonstrating its potential as a powerful invariant for classifying and understanding prime-power groups through computational and theoretical analysis.

Contribution

It introduces the concept of persistent homology for p-groups and coclass trees, and shows its effectiveness as an invariant for small order groups using computational methods.

Findings

Persistent homology provides strong invariants for p-groups of order up to 81.

Theoretical properties suggest usefulness in studying prime-power groups.

Computational evidence supports the potential of persistent homology in group classification.

Abstract

We introduce and investigate notions of persistent homology for p-groups and for coclass trees of p-groups. Using computer techniques we show that persistent homology provides fairly strong homological invariants for p-groups of order at most 81. The strength of these invariants, and some elementary theoretical properties, suggest that persistent homology may be a useful tool in the study of prime-power groups.