gap> ######################### BEGIN COPYRIGHT MESSAGE ######################### GBNP - computing Gröbner bases of noncommutative polynomials Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group at the Department of Mathematics and Computer Science of Eindhoven University of Technology.

For acknowledgements see the manual. The manual can be found in several formats in the doc subdirectory of the GBNP distribution. The acknowledgements formatted as text can be found in the file chap0.txt.

GBNP is free software; you can redistribute it and/or modify it under the terms of the Lesser GNU General Public License as published by the Free Software Foundation (FSF); either version 2.1 of the License, or (at your option) any later version. For details, see the file 'LGPL' in the doc subdirectory of the GBNP distribution or see the FSF's own site: http://www.gnu.org/licenses/lgpl.html gap> ########################## END COPYRIGHT MESSAGE ##########################

gap> ### filename = "example24.g" gap> ### author amc gap> ### amc last edited 31 May 2007

<#GAPDoc Label="Example24">
Serre's exercise
gap> LoadPackage("GBNP","0",false);; gap> SetInfoLevel(InfoGBNP,1); gap> SetInfoLevel(InfoGBNPTime,1);

In Serre's book the following exercise can be found: Prove that the group \langle \{a,b,c\}\mid \{ bab^{-1}a^{-2}, cbc^{-1}b^{-2}, aca^{-1}c^{-2}\}\right\rangle is the trivial group. Here we solve the exercise by running the trace variant of the Gröbner basis function with input the set of equations ba - a^2 b, cb - b^2c, ac - c^2a together with relations forcing that a,b,c are invertible with inverse A,B,C.

gap> KI := [ [[[2,1],[1,1,2]],[1,-1]], > [[[3,2],[2,2,3]],[1,-1]], > [[[1,3],[3,3,1]],[1,-1]], > [[[1,4], []],[1,-1]], > [[[4,1], []],[1,-1]], > [[[2,5], []],[1,-1]], > [[[5,2], []],[1,-1]], > [[[3,6], []],[1,-1]], > [[[6,3], []],[1,-1]], > ];; gap> PrintNPList(KI); ba - a^2b cb - b^2c ac - c^2a ad - 1 da - 1 be - 1 eb - 1 cf - 1 fc - 1

We use , the trace variant of the Gröbner basis computation,

gap> GB := SGrobnerTrace(KI);; #I number of entered polynomials is 9 #I number of polynomials after reduction is 9 #I End of phase I #I End of phase II #I List of todo lengths is [ 9, 10, 11, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 31, 34, 33, 35, 37, 40, 43, 46, 49, 52, 56, 59, 62, 61, 60, 64, 64, 65, 65, 68, 71, 76, 76, 80, 88, 93, 94, 99, 110, 115, 117, 131, 139, 146, 150, 158, 171, 186, 198, 206, 220, 229, 246, 260, 263, 102, 40, 19, 9, 3, 0 ] #I End of phase III #I End of phase IV #I The computation took 19308 msecs.

The dimension of the quotient algebra is 1, showing that the group algebra is 1-dimensional. This implies that the group with the above presentation is trivial. gap> GBpols := List([1..Length(GB)], x -> GB[x].pol);; gap> PrintNPList(GBpols); a - 1 b - 1 c - 1 d - 1 e - 1 f - 1 gap> DimQA(GBpols,6); 1

Since the output is large and might spoil the exercise, we confine ourselves to the printing first polynomial GB[1] and the length of its trace. As the trace polynomial expresses GB[1], which is equal to a-1, as a combination of the binomials in KI, it gives a proof that a can be rewritten within the group to the trivial element. It is easy to derive from this that b and c are also trivial in the group. This justifies the restriction to GB[1].

gap> PrintNP(GB[1].pol); a - 1 gap> Length(GB[1].trace); 1119
<#/GAPDoc>