<html><head><title>[Polycyclic] 6 Libraries and examples of pcp-groups</title></head>
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<h1>6 Libraries and examples of pcp-groups</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP006.htm#SECT001">Libraries of various types of polycyclic groups</a>
<li> <A HREF="CHAP006.htm#SECT002">Some asorted example groups</a>
</ol><p>
<p>
<p>
<h2><a name="SECT001">6.1 Libraries of various types of polycyclic groups</a></h2>
<p><p>
There are the following generic pcp-groups available.
<p>
<a name = "SSEC001.1"></a>
<li><code>AbelianPcpGroup( </code><var>n</var><code>, </code><var>rels</var><code> )</code>
<p>
constructs the abelian group on <var>n</var> generators such that
generator <var>i</var> has order <var>rels[i]</var>. If this order is infinite,
then <var>rels[i]</var> should be either unbound or 0.
<p>
<a name = "SSEC001.2"></a>
<li><code>DihedralPcpGroup( </code><var>n</var><code> )</code>
<p>
constructs the dihedral group of order <var>n</var>. If <var>n</var> is an odd
integer, then 'fail' is returned. If <var>n</var> is zero or not an
integer, then the infinite dihedral group is returned.
<p>
<a name = "SSEC001.3"></a>
<li><code>UnitriangularPcpGroup( </code><var>n</var><code>, </code><var>c</var><code> )</code>
<p>
returns a pcp-group isomorphic to the group of upper triangular
in <var>GL(n, R)</var> where <var>R = <font face="helvetica,arial">Z</font></var> if <var>c = 0</var> and <var>R = <font face="helvetica,arial">F</font><sub>p</sub></var> if <var>c = p</var>.
The natural unitriangular matrix representation of the returned
pcp-group <var>G</var> can be obtained as <var>G!.isomorphism</var>.
<p>
<a name = "SSEC001.4"></a>
<li><code>SubgroupUnitriangularPcpGroup( </code><var>mats</var><code> )</code>
<p>
<var>mats</var> should be a list of upper unitriangular <var>n timesn</var>
matrices over <var><font face="helvetica,arial">Z</font></var> or over <var><font face="helvetica,arial">F</font><sub>p</sub></var>. This function returns the
subgroup of the corresponding 'UnitriangularPcpGroup' generated
by the matrices in <var>mats</var>.
<p>
<a name = "SSEC001.5"></a>
<li><code>InfiniteMetacyclicPcpGroup( </code><var>n</var><code>, </code><var>m</var><code>, </code><var>r</var><code> )</code>
<p>
Infinite metacyclic groups are classified in citeB-K00. Every
infinite metacyclic group <var>G</var> is isomorphic to a finitely presented
group <var>G(m,n,r)</var> with two generators <var>a</var> and <var>b</var> and relations of the
form <var>a<sup>n</sup> = b<sup>m</sup> = 1</var> and <var>[a,b] = a<sup>1-r</sup></var>, where <var>m,n,r</var> are three
non-negative integers with <var>mn=0</var> and <var>r</var> relatively prime to <var>m</var>.
If <var>r equiv-1</var> mod <var>m</var> then <var>n</var> is even, and if <var>r equiv1</var> mod
<var>m</var> then <var>m=0</var>. Also <var>m</var> and <var>n</var> must not be <var>1</var>.
<p>
Moreover, <var>G(m,n,r)congG(m',n',s)</var> if and only if <var>m=m'</var>, <var>n=n'</var>,
and either <var>r equivs</var> or <var>r equivs<sup>-1</sup></var> mod <var>m</var>.
<p>
This function returns the metacyclic group with parameters <var>n</var>,
<var>m</var> and <var>r</var> as a pcp-group with the pc-presentation <var>langle
x,y | x<sup>n</sup>, y<sup>m</sup>, y<sup>x</sup> = y<sup>r</sup>rangle</var>. This presentation is easily
transformed into the one above via the mapping <var>x mapstob<sup>-1</sup>,
y mapstoa</var>.
<p>
<a name = "SSEC001.6"></a>
<li><code>HeisenbergPcpGroup( </code><var>n</var><code> )</code>
<p>
returns the Heisenberg group on 2<strong><var>n</var> generators as pcp-group.
This gives a group of Hirsch length 3</strong><var>n</var>.
<p>
<a name = "SSEC001.7"></a>
<li><code>MaximalOrderByUnitsPcpGroup( </code><var>f</var><code> )</code>
<p>
takes as input a normed, irreducible polynomial over the integers.
Thus <var>f</var> defines a field extension <var>F</var> over the rationals. This
function returns the split extension of the maximal order <var>O</var> of <var>F</var>
by the unit group <var>U</var> of <var>O</var>, where <var>U</var> acts by right multiplication
on <var>O</var>.
<p>
<a name = "SSEC001.8"></a>
<li><code>BurdeGrunewaldPcpGroup( </code><var>s</var><code>, </code><var>t</var><code> )</code>
<p>
returns a nilpotent group of Hirsch length 11 which has been
constructed by Burde und Grunewald. If <var>s</var> is not 0, then this
group has no faithful 12-dimensional linear representation.
<p>
<p>
<h2><a name="SECT002">6.2 Some asorted example groups</a></h2>
<p><p>
The functions in this section provide some more example groups to play
with. They come with no further description and their investigation is
left to the interested user.
<p>
<a name = "SSEC002.1"></a>
<li><code>ExampleOfMetabelianPcpGroup( </code><var>a</var><code>, </code><var>k</var><code> )</code>
<p>
returns an example of a metabelian group. The input parameters must
be two positive integers greater than 1.
<p>
<a name = "SSEC002.2"></a>
<li><code>ExamplesOfSomePcpGroups( </code><var>n</var><code> )</code>
<p>
this function takes values <var>n</var> in 1 up to 16 and returns for each
input an example of a pcp-group. The groups in this example list
have been used as test groups for the functions in this package.
<p>
<p>
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<P>
<address>Polycyclic manual<br>Februar 2009
</address></body></html>
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