Explicit constructions of loops with commuting inner mappings

TL;DR

This paper introduces a method to construct loops with nilpotency class three and abelian inner mappings by modifying group products using symmetric trilinear forms, expanding known examples in loop theory.

Contribution

It provides a new construction technique for loops with specific properties, utilizing central involutions and symmetric trilinear forms, filling a gap in known examples.

Findings

Constructed many such loops from groups of nilpotency class two.

Replaced products with central involutions guided by symmetric trilinear forms.

Extended the class of known loops with abelian inner mappings.

Abstract

In 2004, Cs\"{o}rg\H{o} constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product $xy$ with $xyh$ in certain positions, where $h$ is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.