<Chapter Label="Intro">
<Heading>Introduction</Heading>    

<Section Label="IntroAims">
<Heading>General aims of &Congruence; package</Heading>

<Index Key="IsCongruenceSubgroup"><C>IsCongruenceSubgroup</C></Index>
The &GAP; package &Congruence; provides functions to construct several
types of canonical congruence subgroups in <M>SL_2(&ZZ;)</M>, and also 
intersections of a finite number of such subgroups. 
<P/>

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. 
<P/>

Using the package, one can also determine indices of congruence subgroups and
their intersections in <M>SL_2(&ZZ;)</M> 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.
<P/>

For the theoretical backround, we refer to <Cite Key="LLT-Hecke" />,
<Cite Key="LLT-Algorithm" />, <Cite Key="CLT" /> and <Cite Key="Kulkarni" />.


</Section>

<!-- ********************************************************* -->

<Section Label="IntroInstall">
<Heading>Installation and system requirements</Heading>

&Congruence; is distributed in standard formats (<File>tar.gz</File>, 
<File>tar.bz2</File>, <File>-win.zip</File>) and can be obtained from 
<URL>https://gap-packages.github.io/congruence/</URL>. 
<P/>

&Congruence; does not  use  external  binaries and,  therefore,  works  without 
restrictions on the operating system.  
It requires at least version  &GAP;&nbsp;4.5, and no compatibility with 
previous releases of &GAP;&nbsp;4 is guaranteed.
<P/>

Installation of the package is standard and follows the guidelines from the 
&GAP; manual (see <Ref Sect="Installing a GAP Package" BookName="ref"/>.  
After the package is installed, you can start &GAP; and load 
the &Congruence; package using the command:

<Log>
<![CDATA[
gap> LoadPackage("congruence");
]]>
</Log>



</Section>

</Chapter>
