Sophie: gap-system-4.4.12-5mdv2010.0 i586

gap-system-4.4.12-5mdv2010.0.i586.rpm

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<h1>1 Introduction</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP001.htm#SECT001">Setup for computing the correspondence</a>
<li> <A HREF="CHAP001.htm#SECT002">Collection</a>
</ol><p>
<p>
<a name = "I0"></a>

In this package we demonstrate the algorithmic usefulness of the
so-called Mal'cev correspondence for computations with infinite polycyclic
groups; it is a correspondence
that associates to every <var><font face="helvetica,arial">Q</font></var>-powered nilpotent group <var>H</var> a
unique rational nilpotent Lie algebra <var>L<sub>H</sub></var> and vice-versa.
The Mal'cev correspondence was discovered 
by Anatoly Mal'cev in 1951 <a href="biblio.htm#Mal51"><cite>Mal51</cite></a>.
<p>
<p>
<h2><a name="SECT001">1.1 Setup for computing the correspondence</a></h2>
<p><p>
Let <var>G</var> be a finitely generated torsion-free nilpotent group, 
i.e. a <var>T</var>-group.
Then <var>G</var> can be embedded in a <var><font face="helvetica,arial">Q</font></var>-powered hull <var>hatG</var>.
The group <var>hatG</var> is
a <var><font face="helvetica,arial">Q</font></var>-powered nilpotent group and  
is unique up to isomorphism. 
We denote the Lie algebra
which corresponds to <var>hatG</var> under the Mal'cev correspondence by
<var>L(G)= L<sub>hatG</sub></var>.
<p>
We provide an algorithm for setting up the 
Mal'cev correspondence
between <var>hatG</var> and the Lie algebra <var>L(G)</var>. 
That is, if <var>G</var>
is given by a polycyclic presentation with respect to a Mal'cev basis,
then we can compute a structure constants table of <var>L(G)</var>.
Furthermore for a given <var>ginG</var> we can compute the corresponding 
element in <var>L(G)</var> and vice versa. 
<p>
<p>
<h2><a name="SECT002">1.2 Collection</a></h2>
<p><p>
Every element of a
polycyclically presented
group has a unique normal form. An algorithm for computing this normal
form is called a collection algorithm. Such an algorithm
lies at the heart of most methods
dealing with polycyclically presented groups. The current state of
the art is collection from the left 
citeGeb02,LGS90,VLe90.
<p>
This package contains
a new collection algorithm for polycyclically presented groups,
which we call Mal'cev collection <a href="biblio.htm#ALi07"><cite>ALi07</cite></a>.
Mal'cev collection is
in some cases dramatically faster than
collection from the left, while using less memory.
<p>
<p>
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<address>Example manual<br>June 2007
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