The free commutative automorphic $2$-generated loop of nilpotency class $3$

TL;DR

This paper constructs a specific algebraic structure called the free commutative automorphic 2-generated loop of nilpotency class 3, which has an 8-dimensional integer structure, contributing to the understanding of automorphic loops.

Contribution

It introduces the first explicit construction of the free commutative automorphic 2-generated loop of nilpotency class 3, expanding knowledge of automorphic loop structures.

Findings

Constructed the free commutative automorphic 2-generated loop of nilpotency class 3

Determined the loop's dimension as 8 over the integers

Provided foundational example for further algebraic studies

Abstract

A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic $2$-generated loop of nilpotency class $3$. It has dimension $8$ over the integers.