gap-polycyclic 2.11-3 / usr / share / gap / pkg / Polycyclic / doc / chap6.txt

  
  6 Libraries and examples of pcp-groups
  
  
  6.1 Libraries of various types of polycyclic groups
  
  There are the following generic pcp-groups available.
  
  6.1-1 AbelianPcpGroup
  
  AbelianPcpGroup( n, rels )  function
  
  constructs the abelian group on n generators such that generator i has order
  rels[i]. If this order is infinite, then rels[i] should be either unbound or
  0.
  
  6.1-2 DihedralPcpGroup
  
  DihedralPcpGroup( n )  function
  
  constructs  the  dihedral  group  of  order  n. If n is an odd integer, then
  'fail'  is  returned.  If  n  is  zero  or not an integer, then the infinite
  dihedral group is returned.
  
  6.1-3 UnitriangularPcpGroup
  
  UnitriangularPcpGroup( n, c )  function
  
  returns  a pcp-group isomorphic to the group of upper triangular in GL(n, R)
  where  R = ℤ if c = 0 and R = F_p if c = p. The natural unitriangular matrix
  representation   of   the   returned   pcp-group   G   can  be  obtained  as
  G!.isomorphism.
  
  6.1-4 SubgroupUnitriangularPcpGroup
  
  SubgroupUnitriangularPcpGroup( mats )  function
  
  mats  should  be a list of upper unitriangular n × n matrices over ℤ or over
  F_p.   This   function   returns   the   subgroup   of   the   corresponding
  'UnitriangularPcpGroup' generated by the matrices in mats.
  
  6.1-5 InfiniteMetacyclicPcpGroup
  
  InfiniteMetacyclicPcpGroup( n, m, r )  function
  
  Infinite   metacyclic  groups  are  classified  in  [BK00].  Every  infinite
  metacyclic group G is isomorphic to a finitely presented group G(m,n,r) with
  two  generators  a and b and relations of the form a^n = b^m = 1 and [a,b] =
  a^1-r,  where  m,n,r  are  three  non-negative  integers  with  mn=0  and  r
  relatively  prime  to  m. If r ≡ -1 mod m then n is even, and if r ≡ 1 mod m
  then m=0. Also m and n must not be 1.
  
  Moreover,  G(m,n,r)≅  G(m',n',s) if and only if m=m', n=n', and either r ≡ s
  or r ≡ s^-1 mod m.
  
  This  function  returns the metacyclic group with parameters n, m and r as a
  pcp-group  with  the  pc-presentation  ⟨  x,y  |  x^n, y^m, y^x = y^r⟩. This
  presentation  is  easily  transformed into the one above via the mapping x ↦
  b^-1, y ↦ a.
  
  6.1-6 HeisenbergPcpGroup
  
  HeisenbergPcpGroup( n )  function
  
  returns  the  Heisenberg  group  on 2n generators as pcp-group. This gives a
  group of Hirsch length 3n.
  
  6.1-7 MaximalOrderByUnitsPcpGroup
  
  MaximalOrderByUnitsPcpGroup( f )  function
  
  takes  as  input  a normed, irreducible polynomial over the integers. Thus f
  defines  a  field  extension F over the rationals. This function returns the
  split  extension of the maximal order O of F by the unit group U of O, where
  U acts by right multiplication on O.
  
  6.1-8 BurdeGrunewaldPcpGroup
  
  BurdeGrunewaldPcpGroup( s, t )  function
  
  returns  a nilpotent group of Hirsch length 11 which has been constructed by
  Burde  und  Grunewald.  If  s  is  not  0,  then  this group has no faithful
  12-dimensional linear representation.
  
  
  6.2 Some assorted example groups
  
  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.
  
  6.2-1 ExampleOfMetabelianPcpGroup
  
  ExampleOfMetabelianPcpGroup( a, k )  function
  
  returns  an  example of a metabelian group. The input parameters must be two
  positive integers greater than 1.
  
  6.2-2 ExamplesOfSomePcpGroups
  
  ExamplesOfSomePcpGroups( n )  function
  
  this  function  takes  values  n 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.