Supernilpotent groups and 3-supernilpotent loops

Abstract

We find a short equational basis for the variety of [Formula: see text]-supernilpotent loops. We also present a conceptually simple proof that [Formula: see text]-nilpotence and [Formula: see text]-supernilpotence are equivalent for groups. Connections between [Formula: see text]-supernilpotent loops, Moufang loops, code loops, automorphic loops and AIM loops are explored.