Three concepts of nilpotence in loops

TL;DR

This paper introduces the concept of supernilpotence in loop theory, relating it to existing notions, and shows that finite supernilpotent loops have nilpotent multiplication groups, providing new insights and classical results in the structure of loops.

Contribution

It defines supernilpotence in loops, relates it to existing concepts, and proves finite supernilpotent loops have nilpotent multiplication groups, offering new perspectives in loop theory.

Findings

Supernilpotence class is greater or equal than the class of nilpotence of the multiplication group.

Finite supernilpotent loops have nilpotent multiplication groups.

Loops with nilpotent multiplication groups are centrally nilpotent and admit a prime decomposition.

Abstract

We introduce the abstract concept of supernilpotence in loop theory, and relate it to existing concepts, namely, central nilpotence and nilpotence of the multiplication group. We prove that the class of supernilpotence is greater or equal than the class of nilpotence of the multiplication group, and combining existing results, we show that a finite loop is supernilpotent if and only if its multiplication group is nilpotent. We also provide a new exposition of a classical result and crucial ingredient, that loops with a nilpotent multiplication group are centrally nilpotent and admit a prime decomposition.