A pcp-element is an element of a group defined by a consistent pc-presentation given by a collector. Suppose that g1, ..., gn are the defining generators of the collector. Recall that each element g in this group can be written uniquely as a collected word g1e_1 cdotsgne_n with ei inZ and 0 leqei < ri for i in I. The integer vector [e1, ..., en] is called the exponent vector of g. The following functions can be used to define pcp-elements via their exponent vector or via an arbitrary generator exponent word as introduced in Section Collectors.
PcpElementByExponentsNC( coll, exp )
PcpElementByExponents( coll, exp )
returns the pcp-element with exponent vector exp. The exponent vector is considered relative to the defining generators of the pc-presentation.
PcpElementByGenExpListNC( coll, word )
PcpElementByGenExpList( coll, word )
returns the pcp-element with generators exponent list word. This list word consists of a sequence of generator numbers and their corresponding exponents and is of the form [i1, ei_1, i2, ei_2, ..., ir, ei_r]. The generators exponent list is considered relative to the defining generators of the pc-presentation.
These functions return pcp-elements in the category IsPcpElement.
Presently, the only representation implemented for this category
is IsPcpElementRep.
(This allows us to be a little sloppy right now. The basic set of
operations for IsPcpElement 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.)
IsPcpElement( obj )
returns true if the object obj is a pcp-element.
IsPcpElementRep( obj )
returns true if the object obj is represented as a pcp-element.
Now we can describe attributes and functions for pcp-elements. The
four basic attributes of a pcp-element, Collector, Exponents,
GenExpList and NameTag are computed at the creation of the
pcp-element. All other attributes are determined at runtime.
Let g be a pcp-element and g1, ..., gn a polycyclic generating sequence of the underlying pc-presented group. Let C1, ..., Cn be the polycyclic series defined by g1, ..., gn.
The depth of a non-trivial element g of a pcp-group (with respect to the defining generators) is the integer i such that g inCi setminusCi+1. The depth of the trivial element is defined to be n+1. If gnot=1 has depth i and gie_i cdotsgne_n is the collected word for g, then ei is the leading exponent of g.
If g has depth i, then we call ri = [Ci:Ci+1] the factor order of g. If r<infty, then the smallest positive integer l with gl inCi+1 is the called relative order of g. If r=infty, then the relative order of g is defined to be 0. The index e of langleg,Ci+1rangle in Ci is called relative index of g. We have that r = el.
We call a pcp-element normed, if its leading exponent is equal to its relative index. For each pcp-element g there exists an integer e such that ge is normed.
Collector( g )
the collector to which the pcp-element g belongs.
Exponents( g )
returns the exponent vector of the pcp-element g with respect to the defining generating set of the underlying collector.
GenExpList( g )
returns the generators exponent list of the pcp-element g with respect to the defining generating set of the underlying collector.
NameTag( g )
the name used for printing the pcp-element g. Printing is done by using the name tag and appending the generator number of g.
Depth( g )
returns the depth of the pcp-element g relative to the defining generators.
LeadingExponent( g )
returns the leading exponent of pcp-element g relative to the defining generators. If g is the identity element, the functions returns 'fail'
RelativeOrder( g )
returns the relative order of the pcp-element g with respect to the defining generators.
RelativeIndex( g )
returns the relative index of the pcp-element g with respect to the defining generators.
FactorOrder( g )
returns the factor order of the pcp-element g with respect to the defining generators.
NormingExponent( g )
returns a positive integer e such that the pcp-element g raised to the power of e is normed.
NormedPcpElement( g )
returns the normed element corresponding to the pcp-element g.
labelpcpgroup
A pcp-group is a group consisting of pcp-elements such that all pcp-elements in the group share the same collector. Thus the group G defined by a polycyclic presentation and all its subgroups are pcp-groups.
PcpGroupByCollector( coll )
PcpGroupByCollectorNC( coll )
returns a pcp-group build from the collector coll.
The function calls UpdatePolycyclicCollector (see
UpdatePolycyclicCollector) and checks the confluence (see
IsConfluent) of the collector.
The non-check version bypasses these checks.
Group( gens, id )
returns the group generated by the pcp-elements gens with identity id.
Subgroup( G, gens )
returns a subgroup of the pcp-group G generated by the list gens of pcp-elements from G.
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 ]
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