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Enumeration of Nilpotent Loops by Orbit Counting

Fariha Iftikhar, Gábor P. Nagy

TL;DR

This work reframes the enumeration of nilpotent loops as an explicit orbit-counting problem on the full space of loop cocycles by introducing the affine automorphism group $ ext{Aut}^{}(F,A)$. By showing that orbits on $C(F,A)$ correspond to isomorphism classes of central extensions, the authors apply Burnside’s lemma, reducing the problem to linear-algebra over finite fields, with efficient restrictions via exact and stretched extensions and $p$-tameness. The approach reproduces known classifications for orders $<24$ and provides new counts for order $24$ loops with center size at least $3$, while highlighting remaining limitations for center size $2$ and certain cyclic centers. Practically, the method yields substantial performance gains over prior cohomological enumeration and offers a scalable framework for future extensions and broader orders.

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.

Enumeration of Nilpotent Loops by Orbit Counting

TL;DR

This work reframes the enumeration of nilpotent loops as an explicit orbit-counting problem on the full space of loop cocycles by introducing the affine automorphism group . By showing that orbits on correspond to isomorphism classes of central extensions, the authors apply Burnside’s lemma, reducing the problem to linear-algebra over finite fields, with efficient restrictions via exact and stretched extensions and -tameness. The approach reproduces known classifications for orders and provides new counts for order loops with center size at least , while highlighting remaining limitations for center size and certain cyclic centers. Practically, the method yields substantial performance gains over prior cohomological enumeration and offers a scalable framework for future extensions and broader orders.

Abstract

We study central extensions of nilpotent loops by elementary abelian -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.
Paper Structure (12 sections, 15 theorems, 118 equations, 2 tables)

This paper contains 12 sections, 15 theorems, 118 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Central extensions of loops
  3. Exact and nonexact extensions
  4. Nilpotent loops
  5. Loop cocycles of central extensions
  6. The affine automorphism group and isomorphisms of extensions
  7. Orbit counting by Burnside's lemma
  8. Computing exact extensions
  9. Experimental results on nilpotent loops of order less than 24
  10. Enumeration of nilpotent loops of order 24 with large center
  11. Conclusion
  12. Matrix interpretation of the affine action

Key Result

Lemma 2.2

Let $p$ and $r$ be distinct primes. Fix $[L,A]\in \mathbf{Q}(F,p^d)$ with $F=L/A$. Then $L$ is $r$-stretched if and only if there exists an element $z\in Z_r(F)$ such that $z$ is contained (as a coset) in $Z(L)$.

Theorems & Definitions (36)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • Definition 4.1
  • ...and 26 more