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The ANU Nilpotent Quotient Program
==================================
Nilpotent quotients
-------------------
The lower central series G_i of a group G can be defined inductively
as G_0 = G, G_i = [G_(i-1),G]. G is said to have nilpotency class c if
c is the smallest non-zero integer such that G_c = 1. If N is a normal
subgroup of G and G/N is nilpotent, then N contains G_i for some
non-negative integer i. G has infinite nilpotent quotients if and only
if G/G_1 is infinite. The i-th (i > 1) factor G_(i-1)/G_i of the
lower central series is generated by the elements [g,h]G_i, where g
runs through a set of representatives of G/G_1 and h runs through a
set of representatives of G_(i-2)/G_(i-1).
Any finitely generated nilpotent group is polycyclic and, therefore,
has a subnormal series with cyclic factors. Such a subnormal series
can be used to represent the group in terms of a polycyclic
presentation. The ANU NQ computes successively the factor groups
modulo the terms of the lower central series. Each factor group is
represented by a special form of polycyclic presentation, a nilpotent
presentation, that makes use of the nilpotent structure of the factor
group. Chapters 9 and 11 of the book by C.C. Sims, "Computing with
finitely presented groups", discusses polycyclic presentations and a
nilpotent quotient algorithm. A description of this implementation is
contained in
Werner Nickel (1996) "Computing Nilpotent Quotients of Finitely
Presented Groups" in Dimacs Series in Discrete Mathematics and
Theoretical Computer Science, Volume 25, pp 175-191.
About this version
------------------
This directory contains the Australian National University Nilpotent
Quotient Program (ANU NQ), an implementation of a nilpotent quotient
algorithm in C. This implementation has been developed in a Unix
environment and Unix is currently the only operating system supported.
It runs on a number of different Unix versions. An earlier version of
the ANU NQ is also available as part of quotpic (Derek F. Holt, Sarah
Rees: A graphics system for displaying finite quotients of finitely
presented groups. DIMACS Workshop on Groups and Computation, AMS-ACM
1991).
How to install the ANU NQ
-------------------------
Please refer to the manual for installation instructions.
How to use the ANU NQ
---------------------
Please refer to the manual for instructions on how to use ANU NQ via
the GAP interface or directly via the command line interface.
Acknowledgements
----------------
The author of ANU NQ is Werner Nickel.
The development of this program was started while the author was
supported by an Australian National University PhD scholarship and an
Overseas Postgraduate Research Scholarship.
Further development of this program was done while the author
was supported by the DFG-Schwerpunkt-Projekt "Algorithmische
Zahlentheorie und Algebra".
Since then, maintenance of ANU NQ has been taken over by Max Horn. All
credit for creating ANU NQ still goes to Werner Nickel as sole author.
However, bug reports and other inquiries should be sent to Max Horn.
Contact addresses
-----------------
Bug reports and other requests should be sent to the issue tracker
<https://github.com/gap-packages/nq/issues>
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2026-03-28
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