Construction of groups with triality and their corresponding code loops

TL;DR

This paper generalizes the construction of code loops using groups with triality, demonstrating how to build specific nilpotent groups and Moufang loops with particular properties related to code loops.

Contribution

It introduces a new method for constructing groups with triality and associated Moufang loops, extending previous work to a broader class of loops and groups.

Findings

Constructed nilpotent groups of class 3 with triality

Proved Moufang loops are free loops in a specific variety

Embedded free loops into direct products of smaller loops

Abstract

We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality and $2n$ generators, based on embeddings of $G_n$ into direct products of copies of $G_3$. In the finite case, where $G_n$ is a group such that $|G_n| = 2^{4n+m}$ with $n \ge 3$ and $m = 3 {n \choose 2} + 2 {n \choose 3}$, we prove that the corresponding Moufang loop is the free loop $F_n$ with $n$ generators in the variety generated by code loops. The result depends on a construction similar to that of $G_n$, namely, embedding $F_n$ into direct products of copies of $F_3$, the free code loop associated with $G_3$.