Enumeration of Nilpotent Loops by Orbit Counting

TL;DR

This paper develops an orbit counting method to enumerate nilpotent loops by analyzing central extensions and automorphism group actions, successfully reproducing known results and extending enumeration to order 24.

Contribution

Introduces an affine automorphism group approach for counting isomorphism classes of nilpotent loops via orbit analysis, enabling efficient enumeration.

Findings

Reproduces known nilpotent loops of order less than 24

Enumerates nilpotent loops of order 24 with large centers

Provides a new framework for classifying nilpotent loops

Abstract

We study central extensions of nilpotent loops by elementary abelian $p$-groups using normalized cocycles. By introducing an affine automorphism group acting on the full cocycle space, we obtain a direct correspondence between affine orbits and isomorphism classes of central extensions. This framework yields an efficient orbit-counting method for enumerating nilpotent loops. We reproduce the known results for orders less than 24, and enumerate the nilpotent loops of order 24 with center of size at least 3.