Buchsteiner loops: associators and constructions

TL;DR

This paper investigates the structure of Buchsteiner loops, providing bounds on their order, describing a construction method for certain sizes, and presenting a specific example with particular algebraic properties.

Contribution

It introduces a detailed associator calculus for Buchsteiner loops, establishes order bounds, and offers a construction method for loops of order 32, including a novel example of order 128.

Findings

|Q| ≥ 32 if not conjugacy closed

|Q| ≥ 64 if some element's square is outside the nucleus

Constructed a loop of order 128 with nilpotency class 3

Abstract

Let $Q$ be a Buchsteiner loop. We describe the associator calculus in three variables, and show that $|Q| \ge 32$ if $Q$ is not conjugacy closed. We also show that $|Q| \ge 64$ if there exists $x \in Q$ such that $x^2$ is not in the nucleus of $Q$. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpotency class 3 and possesses an abelian inner mapping group.