Commutators in Central Products of Cayley-Dickson Loops

TL;DR

This paper investigates the behavior of commutators in central products of Cayley-Dickson loops, revealing conditions for triviality and structural isomorphisms, with implications for understanding non-commutative loop constructions.

Contribution

It introduces a sequence of non-commutative loops with high trivial commutator probability and provides a simple proof of isomorphism conditions for central products of Cayley-Dickson loops.

Findings

Probability of trivial commutators approaches 1 in constructed loops

Central product isomorphism implies term-wise isomorphism for n≥3

New insights into the structure of Cayley-Dickson loop products

Abstract

This paper studies the triviality of commutators in central products of Cayley-Dickson loops. Two immediate outcomes of this study are (1) the construction of a sequence of non-commutative loops in which the chance of a random commutator to be trivial approaches 1, and (2) an easy proof for why if two central products of $n$-fold Cayley-Dickson loops are isomorphic for $n\geq 3$, then the loops in the first product are term-wise isomorphic to the loops in the second product.