Supernilpotent groups and $3$-supernilpotent loops

TL;DR

This paper establishes a concise equational basis for 3-supernilpotent loops, proves the equivalence of k-nilpotence and k-supernilpotence in groups, and explores connections among various loop types.

Contribution

It provides a new equational basis for 3-supernilpotent loops and clarifies the relationship between nilpotence and supernilpotence in groups.

Findings

Short equational basis for 3-supernilpotent loops

Proof of equivalence between k-nilpotence and k-supernilpotence in groups

Exploration of connections among different loop classes

Abstract

We find a short equational basis for the variety of $3$-supernilpotent loops. We also present a conceptually simple proof that $k$-nilpotence and $k$-supernilpotence are equivalent for groups. Connections between $3$-supernilpotent loops, Moufang loops, code loops, automorphic loops and AIM loops are explored.