/usr/share/gap/pkg/Polycyclic/doc/chap4.txt

  
  4 Pcp-groups - polycyclically presented groups
  
  
  4.1 Pcp-elements -- elements of a pc-presented group
  
  A   pcp-element   is   an  element  of  a  group  defined  by  a  consistent
  pc-presentation  given  by  a  collector. Suppose that g_1, ..., g_n are the
  defining  generators  of  the  collector. Recall that each element g in this
  group can be written uniquely as a collected word g_1^e_1 ⋯ g_n^e_n with e_i
  ∈  [122ℤX  and  0  ≤  e_i  < r_i for i ∈ I. The integer vector [e_1, ..., e_n] 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 Chapter 3.
  
  4.1-1 PcpElementByExponentsNC
  
  PcpElementByExponentsNC( coll, exp )  function
  PcpElementByExponents( coll, exp )  function
  
  returns  the  pcp-element  with  exponent vector exp. The exponent vector is
  considered relative to the defining generators of the pc-presentation.
  
  4.1-2 PcpElementByGenExpListNC
  
  PcpElementByGenExpListNC( coll, word )  function
  PcpElementByGenExpList( coll, word )  function
  
  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 [i_1, e_i_1, i_2, e_i_2, ..., i_r, e_i_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.)
  
  4.1-3 IsPcpElement
  
  IsPcpElement( obj )  Category
  
  returns true if the object obj is a pcp-element.
  
  4.1-4 IsPcpElementCollection
  
  IsPcpElementCollection( obj )  Category
  
  returns true if the object obj is a collection of pcp-elements.
  
  4.1-5 IsPcpElementRep
  
  IsPcpElementRep( obj )  Representation
  
  returns true if the object obj is represented as a pcp-element.
  
  4.1-6 IsPcpGroup
  
  IsPcpGroup( obj )  Filter
  
  returns true if the object obj is a group and also a pcp-element collection.
  
  
  4.2 Methods for pcp-elements
  
  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 g_1, ..., g_n a polycyclic generating sequence of
  the  underlying  pc-presented  group.  Let  C_1,  ..., C_n be the polycyclic
  series defined by g_1, ..., g_n.
  
  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 ∈ C_i ∖ C_i+1. The depth
  of  the  trivial  element  is  defined  to be n+1. If gnot=1 has depth i and
  g_i^e_i  ⋯  g_n^e_n  is  the  collected  word for g, then e_i is the leading
  exponent of g.
  
  If g has depth i, then we call r_i = [C_i:C_i+1] the factor order of g. If r
  <  [122∞X,  then  the  smallest positive integer l with g^l ∈ C_i+1 is the called
  relative  order  of g. If r=[122∞X, then the relative order of g is defined to be
  0.  The  index e of ⟨ g,C_i+1[122⟩X in C_i 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
  g^e is normed.
  
  4.2-1 Collector
  
  Collector( g )  operation
  
  the collector to which the pcp-element g belongs.
  
  4.2-2 Exponents
  
  Exponents( g )  operation
  
  returns  the  exponent  vector  of  the  pcp-element  g  with respect to the
  defining generating set of the underlying collector.
  
  4.2-3 GenExpList
  
  GenExpList( g )  operation
  
  returns  the  generators  exponent list of the pcp-element g with respect to
  the defining generating set of the underlying collector.
  
  4.2-4 NameTag
  
  NameTag( g )  operation
  
  the  name used for printing the pcp-element g. Printing is done by using the
  name tag and appending the generator number of g.
  
  4.2-5 Depth
  
  Depth( g )  operation
  
  returns the depth of the pcp-element g relative to the defining generators.
  
  4.2-6 LeadingExponent
  
  LeadingExponent( g )  operation
  
  returns  the  leading  exponent  of  pcp-element  g relative to the defining
  generators. If g is the identity element, the functions returns 'fail'
  
  4.2-7 RelativeOrder
  
  RelativeOrder( g )  attribute
  
  returns the relative order of the pcp-element g with respect to the defining
  generators.
  
  4.2-8 RelativeIndex
  
  RelativeIndex( g )  attribute
  
  returns the relative index of the pcp-element g with respect to the defining
  generators.
  
  4.2-9 FactorOrder
  
  FactorOrder( g )  attribute
  
  returns  the  factor order of the pcp-element g with respect to the defining
  generators.
  
  4.2-10 NormingExponent
  
  NormingExponent( g )  function
  
  returns a positive integer e such that the pcp-element g raised to the power
  of e is normed.
  
  4.2-11 NormedPcpElement
  
  NormedPcpElement( g )  function
  
  returns the normed element corresponding to the pcp-element g.
  
  
  4.3 Pcp-groups - groups of pcp-elements
  
  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.
  
  4.3-1 PcpGroupByCollector
  
  PcpGroupByCollector( coll )  function
  PcpGroupByCollectorNC( coll )  function
  
  returns a pcp-group build from the collector coll.
  
  The   function   calls  UpdatePolycyclicCollector  (3.1-6)  and  checks  the
  confluence (see IsConfluent (3.1-7)) of the collector.
  
  The non-check version bypasses these checks.
  
  4.3-2 Group
  
  Group( gens, id )  function
  
  returns the group generated by the pcp-elements gens with identity id.
  
  4.3-3 Subgroup
  
  Subgroup( G, gens )  function
  
  returns  a  subgroup  of  the  pcp-group  G  generated  by  the list gens of
  pcp-elements from G.
  
    Example  
    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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