chap1.txt

  
  1 Introduction
  
  
  1.1 General aims of Congruence package
  
  The  GAP package Congruence provides functions to construct several types of
  canonical  congruence  subgroups  in  SL_2(ℤ),  and  also intersections of a
  finite number of such subgroups.
  
  Furthermore,  it  implements  the algorithm for generating Farey symbols for
  congruence  subgroups  and  using  them  to  produce a system of independent
  generators for these subgroups.
  
  Using  the  package,  one can also determine indices of congruence subgroups
  and  their  intersections  in  SL_2(ℤ)  and  in  other congruence subgroups,
  generate  their  random  elements  and check element memberships. Success of
  other  group  theoretical constructions mostly depends on whether they could
  be expressed in terms of group generators or not.
  
  For  the theoretical backround, we refer to [LLT95b], [LLT95a], [CLLT93] and
  [Kul91].
  
  
  1.2 Installation and system requirements
  
  Congruence  is  distributed  in standard formats (tar.gz, tar.bz2, -win.zip)
  and can be obtained from http://www.cs.st-andrews.ac.uk/~alexk/congruence/.
  
  Congruence  does  not  use  external  binaries and, therefore, works without
  restrictions  on the operating system. It requires at least version GAP 4.5,
  and no compatibility with previous releases of GAP 4 is guaranteed.
  
  Installation  of the package is standard and follows the guidelines from the
  GAP  manual (see 'Reference: Installing a GAP Package'. After the package is
  installed,  you  can  start  GAP  and  load the Congruence package using the
  command:
  
    Example  
    
    gap> LoadPackage("congruence");