Computing with small quasigroups and loops

TL;DR

This paper introduces the GAP package LOOPS for computational algebra, focusing on three problems: loop isomorphisms, classification of small Frattini Moufang loops, and loops with higher nilpotency class.

Contribution

It details the design and capabilities of the LOOPS package and applies it to solve specific computational problems in quasigroup and loop theory.

Findings

Construction of loop isomorphisms demonstrated

Classification of small Frattini Moufang loops of order 64 achieved

Search for loops with higher nilpotency class conducted

Abstract

This is a companion to our lectures GAP and loops, to be delivered at the Workshops Loops 2007, Prague, Czech Republic. In the lectures we introduce the GAP package LOOPS, describe its capabilities, and explain in detail how to use it. In this paper we first outline the philosophy behind the package and its main features, and then we focus on three particular computational problems: construction of loop isomorphisms, classification of small Frattini Moufang loops of order 64, and the search for loops of nilpotency class higher than two with an abelian inner mapping group. In particular, this is not a user's manual for LOOPS, which can be downloaded from the distribution website of LOOPS.