Construction of Co₃. An example of the use of an integrated system for computational group theory

Summary

Introduction

This paper aims to demonstrate, by example, a small sample of the capabilities of the GAP system [S⁺97] for computational algebra. We specifically focus on the advantages arising from the use of an integrated system such as GAP, which allows the easy combination of techniques from a range of areas, without requiring the user to have a detailed knowledge of the algorithms used.

The sporadic group Co₃ has a faithful permutation representation on 276 points which is unusually small for a group of its size. We want to construct this permutation representation by way of a chain of subgroups of ascending order. In this process we will construct explicitly the sporadic simple groups M₂₂ and HS together with associated graphs and codes. Our guide in this is the ATLAS of Finite Simple Groups [CCN⁺85], which contains a variety of information about the groups of interest, including very terse “constructions” – outlines of settings in which these groups can occur.

We will use GAP (version 3.4, patchlevel 4, including the GRAPE [Soi93] and GUAVA [BCMR] share packages) to realise these constructions. We will see that the integration of many algorithms in one system will permit us to follow the path outlined in theory with concrete constructions.

From a computational stand-point, this is not a very large or difficult computation. It is interesting, however, because it uses a very wide range of techniques, and because it demonstrates one important way in which an integrated system such as GAP can be used.