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<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 < 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>
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<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 < \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>.
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