The structure of extra loops

TL;DR

This paper explores the algebraic structure of finite and infinite extra loops, establishing key theorems, properties of centers, and classifications of loops of certain orders, advancing understanding of their algebraic behavior.

Contribution

It proves Sylow and P. Hall theorems for finite extra loops, analyzes centers and quotient structures, and classifies nonassociative extra loops of specific orders.

Findings

Sylow theorems hold for finite extra loops

Finite nonassociative extra loops have nontrivial centers

Exactly 16 nonassociative extra loops of order 16p for each odd prime p

Abstract

The Sylow theorems hold for finite extra loops, as does P. Hall's theorem for finite solvable extra loops. Every finite nonassociative extra loop $Q$ has a nontrivial center, $Z(Q)$. Furthermore, $Q/Z(Q)$ is a group whenever $|Q| < 512$. Loop extensions are used to construct an infinite nonassociative extra loop with a trivial center and a nonassociative extra loop $Q$ of order 512 such that $Q/Z(Q)$ is nonassociative. There are exactly 16 nonassociative extra loops of order $16p$ for each odd prime $p$.