1 2The ANU Nilpotent Quotient Program 3================================== 4 5 6Nilpotent quotients 7------------------- 8 9The lower central series G_i of a group G can be defined inductively 10as G_0 = G, G_i = [G_(i-1),G]. G is said to have nilpotency class c if 11c is the smallest non-zero integer such that G_c = 1. If N is a normal 12subgroup of G and G/N is nilpotent, then N contains G_i for some 13non-negative integer i. G has infinite nilpotent quotients if and only 14if G/G_1 is infinite. The i-th (i > 1) factor G_(i-1)/G_i of the 15lower central series is generated by the elements [g,h]G_i, where g 16runs through a set of representatives of G/G_1 and h runs through a 17set of representatives of G_(i-2)/G_(i-1). 18 19Any finitely generated nilpotent group is polycyclic and, therefore, 20has a subnormal series with cyclic factors. Such a subnormal series 21can be used to represent the group in terms of a polycyclic 22presentation. The ANU NQ computes successively the factor groups 23modulo the terms of the lower central series. Each factor group is 24represented by a special form of polycyclic presentation, a nilpotent 25presentation, that makes use of the nilpotent structure of the factor 26group. Chapters 9 and 11 of the book by C.C. Sims, "Computing with 27finitely presented groups", discusses polycyclic presentations and a 28nilpotent quotient algorithm. A description of this implementation is 29contained in 30 31Werner Nickel (1996) "Computing Nilpotent Quotients of Finitely 32Presented Groups" in Dimacs Series in Discrete Mathematics and 33Theoretical Computer Science, Volume 25, pp 175-191. 34 35 36About this version 37------------------ 38 39This directory contains the Australian National University Nilpotent 40Quotient Program (ANU NQ), an implementation of a nilpotent quotient 41algorithm in C. This implementation has been developed in a Unix 42environment and Unix is currently the only operating system supported. 43It runs on a number of different Unix versions. An earlier version of 44the ANU NQ is also available as part of quotpic (Derek F. Holt, Sarah 45Rees: A graphics system for displaying finite quotients of finitely 46presented groups. DIMACS Workshop on Groups and Computation, AMS-ACM 471991). 48 49 50How to install the ANU NQ 51------------------------- 52 53Please refer to the manual for installation instructions. 54 55How to use the ANU NQ 56--------------------- 57 58Please refer to the manual for instructions on how to use ANU NQ via 59the GAP interface or directly via the command line interface. 60 61 62Acknowledgements 63---------------- 64The author of ANU NQ is Werner Nickel. 65 66The development of this program was started while the author was 67supported by an Australian National University PhD scholarship and an 68Overseas Postgraduate Research Scholarship. 69 70Further development of this program was done while the author 71was supported by the DFG-Schwerpunkt-Projekt "`Algorithmische 72Zahlentheorie und Algebra"'. 73 74Since then, maintenance of ANU NQ has been taken over by Max Horn. All 75credit for creating ANU NQ still goes to Werner Nickel as sole author. 76However, bug reports and other inquiries should be sent to Max Horn. 77 78 79Contact addresses 80----------------- 81Bug reports and other requests should be sent to the issue tracker 82 83 https://github.com/gap-packages/nq/issues 84