Admissible orders of Jordan loops

TL;DR

This paper characterizes the possible orders of nonassociative Jordan loops, proving existence for all orders ≥6 except 9, and constructs an infinite family of finite simple nonassociative Jordan loops.

Contribution

It provides a complete classification of the orders of nonassociative Jordan loops and introduces new constructions for simple Jordan loops.

Findings

Nonassociative Jordan loops exist if and only if order ≥6 and not 9.

Constructed an infinite family of finite simple nonassociative Jordan loops.

Analyzed powers of elements in Jordan loops and their well-definedness.

Abstract

A commutative loop is Jordan if it satisfies the identity $x^2 (y x) = (x^2 y) x$. Using an amalgam construction and its generalizations, we prove that a nonassociative Jordan loop of order $n$ exists if and only if $n\geq 6$ and $n\neq 9$. We also consider whether powers of elements in Jordan loops are well-defined, and we construct an infinite family of finite simple nonassociative Jordan loops.