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<Chapter Label="Libraries and examples of pcp-groups">
<Heading>Libraries and examples of pcp-groups</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Libraries of various types of polycyclic groups">
<Heading>Libraries of various types of polycyclic groups</Heading>
There are the following generic pcp-groups available.
<ManSection>
<Func Name="AbelianPcpGroup" Arg="n, rels"/>
<Description>
constructs the abelian group on <A>n</A> generators such that
generator <M>i</M> has order <M>rels[i]</M>. If this order is infinite,
then <M>rels[i]</M> should be either unbound or 0.
</Description>
</ManSection>
<ManSection>
<Func Name="DihedralPcpGroup" Arg="n"/>
<Description>
constructs the dihedral group of order <A>n</A>. If <A>n</A> is an odd
integer, then 'fail' is returned. If <A>n</A> is zero or not an
integer, then the infinite dihedral group is returned.
</Description>
</ManSection>
<ManSection>
<Func Name="UnitriangularPcpGroup" Arg="n, c"/>
<Description>
returns a pcp-group isomorphic to the group of upper triangular
in <M>GL(n, R)</M> where <M>R = &ZZ;</M> if <M>c = 0</M> and <M>R = \mathbb{F}_p</M> if <M>c = p</M>.
The natural unitriangular matrix representation of the returned
pcp-group <M>G</M> can be obtained as <M>G!.isomorphism</M>.
</Description>
</ManSection>
<ManSection>
<Func Name="SubgroupUnitriangularPcpGroup" Arg="mats"/>
<Description>
<A>mats</A> should be a list of upper unitriangular <M>n \times n</M>
matrices over <M>&ZZ;</M> or over <M>\mathbb{F}_p</M>. This function returns the
subgroup of the corresponding 'UnitriangularPcpGroup' generated
by the matrices in <A>mats</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="InfiniteMetacyclicPcpGroup" Arg="n, m, r"/>
<Description>
Infinite metacyclic groups are classified in <Cite Key="B-K00"/>. Every
infinite metacyclic group <M>G</M> is isomorphic to a finitely presented
group <M>G(m,n,r)</M> with two generators <M>a</M> and <M>b</M> and relations of the
form <M>a^n = b^m = 1</M> and <M>[a,b] = a^{1-r}</M>, where <M>m,n,r</M> are three
non-negative integers with <M>mn=0</M> and <M>r</M> relatively prime to <M>m</M>.
If <M>r \equiv -1</M> mod <M>m</M> then <M>n</M> is even, and if <M>r \equiv 1</M> mod
<M>m</M> then <M>m=0</M>. Also <M>m</M> and <M>n</M> must not be <M>1</M>.
<P/>
Moreover, <M>G(m,n,r)\cong G(m',n',s)</M> if and only if <M>m=m'</M>, <M>n=n'</M>,
and either <M>r \equiv s</M> or <M>r \equiv s^{-1}</M> mod <M>m</M>.
<P/>
This function returns the metacyclic group with parameters <A>n</A>,
<A>m</A> and <A>r</A> as a pcp-group with the pc-presentation <M>\langle
x,y | x^n, y^m, y^x = y^r\rangle</M>. This presentation is easily
transformed into the one above via the mapping <M>x \mapsto b^{-1},
y \mapsto a</M>.
</Description>
</ManSection>
<ManSection>
<Func Name="HeisenbergPcpGroup" Arg="n"/>
<Description>
returns the Heisenberg group on <M>2<A>n</A></M> generators as pcp-group.
This gives a group of Hirsch length <M>3<A>n</A></M>.
</Description>
</ManSection>
<ManSection>
<Func Name="MaximalOrderByUnitsPcpGroup" Arg="f"/>
<Description>
takes as input a normed, irreducible polynomial over the integers.
Thus <A>f</A> defines a field extension <A>F</A> over the rationals. This
function returns the split extension of the maximal order <A>O</A> of <A>F</A>
by the unit group <A>U</A> of <A>O</A>, where <A>U</A> acts by right multiplication
on <A>O</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="BurdeGrunewaldPcpGroup" Arg="s, t"/>
<Description>
returns a nilpotent group of Hirsch length 11 which has been
constructed by Burde und Grunewald. If <A>s</A> is not 0, then this
group has no faithful 12-dimensional linear representation.
</Description>
</ManSection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Some asorted example groups">
<Heading>Some assorted example groups</Heading>
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.
<ManSection>
<Func Name="ExampleOfMetabelianPcpGroup" Arg="a, k"/>
<Description>
returns an example of a metabelian group. The input parameters must
be two positive integers greater than 1.
</Description>
</ManSection>
<ManSection>
<Func Name="ExamplesOfSomePcpGroups" Arg="n"/>
<Description>
this function takes values <A>n</A> 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.
</Description>
</ManSection>
</Section>
</Chapter>
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