Powers of Elements in Jordan Loops

TL;DR

This paper investigates the properties of powers in Jordan loops, proving key identities and demonstrating the nonexistence of nonassociative Jordan loops of order 9.

Contribution

It establishes new identities involving powers in Jordan loops and proves the nonexistence of certain small nonassociative Jordan loops.

Findings

Derived several identities involving powers in Jordan loops.

Proved that no nonassociative Jordan loop of order 9 exists.

Abstract

A Jordan loop is a commutative loop satisfying the Jordan identity $(x^2 y) x = x^2 (y x)$. We establish several identities involving powers in Jordan loops and show that there is no nonassociative Jordan loop of order $9$.