Enumeration of Nilpotent Loops by Orbit Counting

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.

💡 Research Summary

The paper tackles the long‑standing problem of classifying finite nilpotent loops up to isomorphism by exploiting the theory of central extensions and normalized 2‑cocycles. Let F be a finite loop and A an elementary abelian p‑group (viewed additively). Any central extension L of F by A can be identified, after fixing a transversal, with the set F × A and its multiplication is completely determined by a normalized cocycle θ ∈ C(F,A) via
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