<Chapter Label="Pcp-groups - polycyclically presented groups">
<Heading>Pcp-groups - polycyclically presented groups</Heading>

<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Pcp-elements -- elements of a pc-presented group">
<Heading>Pcp-elements -- elements of a pc-presented group</Heading>

A <E>pcp-element</E> is an element of a group defined by a consistent
pc-presentation given by a collector. Suppose that <M>g_1, \ldots, g_n</M>
are the defining generators of the collector. Recall that each element
<M>g</M> in this group can be written uniquely as a collected word <M>g_1^{e_1}
\cdots g_n^{e_n}</M> with <M>e_i \in &ZZ;</M> and <M>0 \leq e_i &lt; r_i</M> for <M>i \in
I</M>. The integer vector <M>[e_1, \ldots, e_n]</M> is called the <E>exponent
vector</E> of <M>g</M>.  The following functions can be used to define
pcp-elements via their exponent vector or via an arbitrary generator
exponent word as introduced in Chapter <Ref Chap="Collectors"/>.

<ManSection>
<Func Name="PcpElementByExponentsNC" Arg="coll, exp"/>
<Func Name="PcpElementByExponents" Arg="coll, exp"/>
<Description>
	returns the pcp-element with exponent vector <A>exp</A>. The exponent vector
	is considered relative to the defining generators of the pc-presentation.
</Description>
</ManSection>

<ManSection>
<Func Name="PcpElementByGenExpListNC" Arg="coll, word"/>
<Func Name="PcpElementByGenExpList" Arg="coll, word"/>
<Description>
	returns the pcp-element with generators exponent list <A>word</A>. This list
	<A>word</A> consists of a sequence of generator numbers and their corresponding
	exponents and is of the form <M>[i_1, e_{i_1}, i_2, e_{i_2}, \ldots, i_r,
	e_{i_r}]</M>. The
	generators exponent list is considered relative to the defining generators
	of the pc-presentation.
	<P/>

	These functions return pcp-elements in the category <C>IsPcpElement</C>.
	Presently,  the  only representation  implemented for this category
	is <C>IsPcpElementRep</C>.
	(This allows us  to be a  little sloppy right now.  The basic  set of
	operations for  <C>IsPcpElement</C> has  not been defined yet.  This is
	going to happen in one of the next version, certainly as soon as the
	need for different representations arises.)
</Description>
</ManSection>

<ManSection>
<Filt Name="IsPcpElement" Arg="obj" Type='Category'/>
<Description>
	returns true if the object <A>obj</A> is a pcp-element.
</Description>
</ManSection>

<ManSection>
<Filt Name="IsPcpElementCollection" Arg="obj" Type='Category'/>
<Description>
	returns true if the object <A>obj</A> is a collection of pcp-elements.
</Description>
</ManSection>

<ManSection>
<Filt Name="IsPcpElementRep" Arg="obj" Type='Representation'/>
<Description>
	returns true if the object <A>obj</A> is represented as a pcp-element.
</Description>
</ManSection>

<ManSection>
<Filt Name="IsPcpGroup" Arg="obj" Type='Filter'/>
<Description>
	returns true if the object <A>obj</A> is a group 
        and also a pcp-element collection.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Methods for pcp-elements">
<Heading>Methods for pcp-elements</Heading>

Now we can describe attributes and functions for pcp-elements. The
four basic attributes of a pcp-element, <C>Collector</C>, <C>Exponents</C>,
<C>GenExpList</C> and <C>NameTag</C> are computed at the creation of the
pcp-element. All other attributes are determined at runtime.
<P/>

Let <A>g</A> be a pcp-element and <M>g_1, \ldots, g_n</M> a polycyclic generating
sequence of the underlying pc-presented group. Let <M>C_1, \ldots, C_n</M>
be the polycyclic series defined by <M>g_1, \ldots, g_n</M>.
<P/>

The <E>depth</E> of a non-trivial element <M>g</M> of  a pcp-group (with respect
to the defining generators) is the integer <M>i</M> such that <M>g \in C_i
\setminus C_{i+1}</M>. The depth  of the trivial element is defined to
be <M>n+1</M>. If <M>g\not=1</M> has depth <M>i</M> and <M>g_i^{e_i} \cdots g_n^{e_n}</M>
is the collected word for <M>g</M>, then <M>e_i</M> is the <E>leading exponent</E> of
<M>g</M>.
<P/>

If  <M>g</M> has  depth <M>i</M>, then we call <M>r_i = [C_i:C_{i+1}]</M> the <E>factor
order</E> of <M>g</M>. If <M>r &lt; \infty</M>, then the  smallest positive integer <M>l</M>
with <M>g^l  \in C_{i+1}</M>  is the called  <E>relative  order</E> of  <M>g</M>.  If
<M>r=\infty</M>, then the relative order  of <M>g</M> is defined  to be <M>0</M>. The
index <M>e</M>   of <M>\langle g,C_{i+1}\rangle</M>  in  <M>C_i</M> is called <E>relative
index</E>   of  <M>g</M>.   We   have that   <M>r  =  el</M>.
<P/>

We  call a pcp-element <E>normed</E>, if its leading  exponent is equal to
its relative index. For  each pcp-element <M>g</M>  there exists an integer
<M>e</M>  such   that <M>g^e</M>  is  normed.

<ManSection>
<Oper Name="Collector" Arg="g"/>
<Description>
	the collector to  which the pcp-element  <A>g</A> belongs.
</Description>
</ManSection>

<ManSection>
<Oper Name="Exponents" Arg="g"/>
<Description>
	returns the exponent vector of the pcp-element <A>g</A> with respect to the defining
	generating set of the underlying collector.
</Description>
</ManSection>

<ManSection>
<Oper Name="GenExpList" Arg="g"/>
<Description>
	returns the generators  exponent  list  of the  pcp-element  <A>g</A> with respect to
	the defining generating set of the underlying collector.
</Description>
</ManSection>

<ManSection>
<Oper Name="NameTag" Arg="g"/>
<Description>
	the name used for  printing the pcp-element <A>g</A>.   Printing is done by
	using the name tag and appending the generator number of <A>g</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="Depth" Arg="g"/>
<Description>
	returns  the  depth of the  pcp-element  <A>g</A> relative to  the defining
	generators.
</Description>
</ManSection>

<ManSection>
<Oper Name="LeadingExponent" Arg="g"/>
<Description>
	returns  the  leading exponent  of  pcp-element  <A>g</A>  relative to  the
	defining generators.  If  <A>g</A> is the  identity element, the  functions
	returns 'fail'
</Description>
</ManSection>

<ManSection>
<Attr Name="RelativeOrder" Arg="g"/>
<Description>
	returns the relative order of the  pcp-element <A>g</A> with respect to the
	defining generators.
</Description>
</ManSection>

<ManSection>
<Attr Name="RelativeIndex" Arg="g"/>
<Description>
	returns the relative index of the pcp-element <A>g</A>  with respect to the
	defining generators.
</Description>
</ManSection>

<ManSection>
<Attr Name="FactorOrder" Arg="g"/>
<Description>
	returns  the factor order  of the pcp-element <A>g</A>  with respect to the
	defining generators.
</Description>
</ManSection>

<ManSection>
<Func Name="NormingExponent" Arg="g"/>
<Description>
	returns a positive integer <M>e</M> such that the pcp-element <A>g</A> raised to
	the power of <M>e</M> is normed.
</Description>
</ManSection>

<ManSection>
<Func Name="NormedPcpElement" Arg="g"/>
<Description>
	returns the normed element corresponding to the pcp-element <A>g</A>.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="pcpgroup">
<Heading>Pcp-groups - groups of pcp-elements</Heading>

A  <E>pcp-group</E> is a  group consisting of pcp-elements such that all
pcp-elements in  the group share  the same collector. Thus the group
<M>G</M>  defined by a polycyclic presentation and all its subgroups are
pcp-groups.

<ManSection>
<Func Name="PcpGroupByCollector" Arg="coll"/>
<Func Name="PcpGroupByCollectorNC" Arg="coll"/>
<Description>
	returns a pcp-group build from the collector <A>coll</A>.
	<P/>

	The function calls <Ref Func="UpdatePolycyclicCollector"/>
	 and checks the confluence (see
	<Ref Func="IsConfluent"/>) of the collector.
	<P/>

	The non-check version bypasses these checks.
</Description>
</ManSection>

<ManSection>
<Func Name="Group" Arg="gens, id"/>
<Description>
	returns the group generated by the pcp-elements <A>gens</A> with identity
	<A>id</A>.
</Description>
</ManSection>

<ManSection>
<Func Name="Subgroup" Arg="G, gens"/>
<Description>
	returns a subgroup of the pcp-group <A>G</A> generated by the list <A>gens</A> of
	pcp-elements from <A>G</A>.

<Example><![CDATA[
gap>  ftl := FromTheLeftCollector( 2 );;
gap>  SetRelativeOrder( ftl, 1, 2 );
gap>  SetConjugate( ftl, 2, 1, [2,-1] );
gap>  UpdatePolycyclicCollector( ftl );
gap>  G:= PcpGroupByCollectorNC( ftl );
Pcp-group with orders [ 2, 0 ]
gap> Subgroup( G, GeneratorsOfGroup(G){[2]} );
Pcp-group with orders [ 0 ]
]]></Example>
</Description>
</ManSection>

</Section>
</Chapter>