Quelle example24.xml Sprache: XML

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:
https://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">
<Section Label="Example24"><Heading>Serre's exercise

<Listing><![CDATA[
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,1);
]]></Listing>

In Serre's book the following exercise can be found:
Prove  that the group
<M> \left\langle \{a,b,c\}\mid \{ bab^{-1}a^{-2}, cbc^{-1}b^{-2}, aca^{-1}c^{-2}\}\right\rangle</M>
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
<M>ba - a^2 b, cb - b^2c, ac - c^2a</M>
together with relations forcing that <M>a,b,c</M> are invertible
with inverse <M>A,B,C</M>.

<Listing><![CDATA[
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
]]></Listing>

We use <Ref Func="SGrobnerTrace" Style="Text"/>,
the trace variant of the Gröbner basis computation,

<Listing><![CDATA[
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.
]]></Listing>

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.
<Listing><![CDATA[
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
]]></Listing>

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

<Listing><![CDATA[
gap> PrintNP(GB[1].pol);
a - 1
gap> Length(GB[1].trace);
1119
]]></Listing>

</Section>
<#/GAPDoc>

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