Enumeration of nilpotent loops via cohomology

TL;DR

This paper develops cohomology-based tools to classify and enumerate nilpotent loops, providing both theoretical counting methods and practical algorithms, successfully enumerating all such loops of order less than 24.

Contribution

It introduces cohomology and linear algebra techniques for classifying nilpotent loops, enabling enumeration and efficient computer classification.

Findings

Enumerated all nilpotent loops of order less than 24.

Developed cohomology-based methods for counting isomorphism classes.

Created algorithms for classifying nilpotent loops efficiently.

Abstract

The isomorphism problem for centrally nilpotent loops can be tackled by methods of cohomology. We develop tools based on cohomology and linear algebra that either lend themselves to direct count of the isomorphism classes (notably in the case of nilpotent loops of order $2q$, $q$ a prime), or lead to efficient classification computer programs. This allows us to enumerate all nilpotent loops of order less than $24$.