Buchsteiner loops

TL;DR

This paper investigates Buchsteiner loops, demonstrating that their quotient by the nucleus forms an abelian group of exponent four, and provides an explicit example where this exponent is realized.

Contribution

It establishes the structure of Buchsteiner loops modulo their nucleus and constructs a specific example illustrating the theoretical result.

Findings

Buchsteiner loops modulo their nucleus are abelian groups of exponent four.

An explicit example of a Buchsteiner loop with the quotient of exponent four.

The identity defining Buchsteiner loops leads to this structural characterization.

Abstract

Buchsteiner loops are those which satisfy the identity $x\backslash (xy \cdot z) = (y \cdot zx)/ x$. We show that a Buchsteiner loop modulo its nucleus is an abelian group of exponent four, and construct an example where the factor achieves this exponent.