How Associative Can a Non-Associative Moufang Loop Be?

TL;DR

This paper extends the classical $\frac{5}{8}$-theorem to two non-associative settings by analyzing the probability that three randomly chosen elements associate in finite Moufang loops with nuclear commutators and in finite non-associative CC loops. For Moufang loops with $[G,G,[G,G]]=1$, the probability is bounded above by $\frac{43}{64}$, and this bound is tight, realized by the octonion loop $${\mathbb{O}}_{16}$$; for CC loops the bound is $\frac{7}{8}$, with sharpness evident in the smallest CC loop of order $6$. The proofs decompose the associativity event into cases based on nucleus membership and adjoint-sets, using Bruck's adjoint construction and subloop-size constraints to derive the probabilistic limits. The results shed light on structural restrictions that force associativity and provide computational verification for the extremal examples, while suggesting potential limitations of such bounds beyond the two studied classes.

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.

Theorems & Definitions (17)

• Theorem
• Theorem
• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3