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).

Nilmat manual

September 2017