Simple Moufang loops and Galois extensions

TL;DR

This paper reveals a one-to-one correspondence between simple Moufang loops and Paige loops over Galois extensions, providing a comprehensive description of their structure, automorphisms, and subloops.

Contribution

It establishes a novel connection between simple Moufang loops and Galois extensions, fully characterizing nonassociative finite simple Moufang loops.

Findings

Describes generators and subloop structures

Determines automorphism groups of these loops

Classifies all nonassociative finite simple Moufang loops

Abstract

The paper establishes an one-to-one correspondence between simple Moufang loops and Paige loops constructed over Galois extension over prime field in its algebraic closure. Using this connection it describes fully the family of nonassociative finite simple Moufang loops.It describes the generators, the loop structure of subloops, the automorphism group of nonassociative simple Moufang loops.