Admissible orders of Jordan loops

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.