Construction of a Family of Nafil Loops of Odd Order n = 2m +1

TL;DR

This paper constructs an infinite family of non-associative finite invertible loops (NAFIL) of every odd order n ≥ 5, highlighting their algebraic properties and potential applications in physics.

Contribution

It introduces a new family of NAFIL loops of all odd orders n ≥ 5, expanding the known classes of such algebraic structures.

Findings

Existence of NAFIL loops for all odd n ≥ 5 established

The first NAFIL loop of order 5 relates to Lie algebra applications

The family is simple, power-associative, and non-associative

Abstract

The existence of NAFIL loops of every odd order n => 5 is established by construction. These are non-associative finite invertible loops that are simple and power-associative and they form an infinite family. The first member of this family is the NAFIL loop of order n = 5 which is known to define a Lie algebra with some possible applications in particle physics.