On the uniqueness of loops M(G,2)

TL;DR

This paper investigates the construction of Moufang loops called M(G,2) derived from a finite group G and explores the conditions under which these loops are non-associative, focusing on their uniqueness and structural properties.

Contribution

It characterizes the specific assignments of multiplicative operations on G×C2 that produce non-associative Moufang loops M(G,2) and establishes their uniqueness up to isomorphism.

Findings

Exactly four assignments produce non-associative Moufang loops for nonabelian G.

All such loops are isomorphic, showing a form of uniqueness.

The loops M(G,2) are characterized by specific multiplicative operation assignments.

Abstract

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i$, $j$, $k\in\{-1, 1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times\{i\})\times(G\times\{j\})$, for $i$, $j\in C_2$. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.