How Associative Can a Non-Associative Moufang Loop Be?

TL;DR

This paper establishes a new probabilistic criterion for finite Moufang loops with nuclear commutators, showing that high associativity probability implies the loop is actually a group, extending classical associativity results.

Contribution

It provides a non-associative analog to the 5/8 Theorem, identifying a precise probability threshold that guarantees a Moufang loop is associative.

Findings

If the associativity probability exceeds 43/64, the loop is a group.

The bound of 43/64 is proven to be tight with the 16-element Octonion loop.

The result applies specifically to finite Moufang loops with nuclear commutators.

Abstract

We prove a non-associative analog to the well-known $\frac{5}{8}$ Theorem. Namely, for a finite Moufang loop with nuclear commutators, we show that if the probability that three randomly chosen elements associate is greater than $\frac{43}{64}$, then the loop must be a group. The bound is tight as demonstrated by the 16-element Octonion loop.