Nilpotent Quotients of Finitely Presented Groups

Version 2.5.8
Released 2022-04-04

This project is maintained by Max Horn

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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.


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