[Nilmat] 1 Introduction

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# 1 Introduction

This package is for computing with nilpotent matrix groups over a field F, where F is a finite field GF(q) or the rational number field Q.

Nilmat contains an implementation of algorithms developed over the past few years, available in theoretical form in the papers DF04,DF05b,DF06,DF07. The theory of nilpotent matrix groups is an essential part of linear group theory. Many structural and classification results for nilpotent linear groups are known (see e.g. Sup76,Weh73), and specialized methods for handling these groups have been developed. The computational advantages of nilpotent linear groups have been addressed in DF05a. For a full description of most of the algorithms of this package, further general information, and historical remarks, see DF06,DF07.

One purpose of Nilmat is testing nilpotency of a subgroup G of GL(n,F). If G < GL(n,Q) is found to be nilpotent then the package provides a function for deciding whether G is finite. If G < GL(n,q) is found to be nilpotent then the package provides a function that returns the Sylow subgroups of G. Additional functions allow one to test whether a nilpotent subgroup G of GL(n,F) is completely reducible or unipotent, and to compute the order of G if it is finite.

Another feature of Nilmat is a library of nilpotent primitive matrix groups. Specifically, for each integer n > 1 and prime power q, this library returns a complete and irredundant list of GL(n,q)-conjugacy class representatives of the nilpotent primitive subgroups of GL(n,q).

The problem of constructing nilpotent matrix groups is interesting in its own right. We have included in the package functions concerned with this problem. For example, one such function constructs maximal absolutely irreducible nilpotent subgroups of GL(n,q).

Related research on solvable and polycyclic matrix groups was carried out by Björn Assmann and Bettina Eick in AE05,AE07. Most of the algorithms in AE05 were implemented in the GAP package Polenta, on which Nilmat partially relies.

This work has emanated from research conducted with the financial support of Science Foundation Ireland and the German Academic Exchange Service (DAAD).

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