<!-- This is an automatically generated file. -->
<Chapter Label="Chapter_Introduction">
<Heading>Introduction</Heading>
<P/>
The &corefreesub; package was created to calculate core-free subgroups of a group, their indexes, and faithful transitive permutation representations.
<P/>
A core-free subgroup of a group <A>G</A> is a subgroup <A>H</A> such that
<Display> \cap_{g\in G} H = \{id_G\}. </Display>
These subgroups are important since the action of <A>G</A> on the cosets of <A>H</A> is both transitive and faithful.
Hence, this gives us a faithful transitive permutation representation of <A>G</A> with degree <A>n</A>, where <A>n</A> is the index of <A>H</A> in <A>G</A>.
<P/>
There are many articles studying faithful permutation representation of groups, such as <Cite Key="johnson_minimal_1971"/>, <Cite Key="easdown_minimal_1988"/>, <Cite Key="saunders_minimal_2014"/> and <Cite Key="easdown_minimal_2016"/>.
However the restriction on transitive actions is more recent and there are fewer studies like <Cite Key="FP20Tor"/>,<Cite Key="FP21Cor"/>,<Cite Key="FP21Hyper"/> and <Cite Key="FP22Loc"/>.
<P/>
During C.A. Piedade's PhD thesis, he studied many of these faithful transitive permutation representations of automorphism groups of abstract regular polytopes and hypertopes.
It was also during this period that this author noticed the absence of functions/methods in GAP to compute core-free subgroups of a group.
As a consequence, he created many functions to help in his research, resulting in many of the functions and methods implemented in this package.
<P/>
One of the important tools for studying faithful transitive permutation representations is by using <A>faithful transitive permutation representation graphs</A>, which are <A>Schreier coset graphs</A>.
A <A>Schreier coset graph</A> is a graph associated with a group <A>G</A>, its generators and a subgroup <A>H</A> of <A>G</A>.
The vertices of the graph are the right cosets of <A>H</A> and there is a directed edge <Math>(Hx,Hy)</Math>
with label <Math>g</Math> if <Math>g</Math> is a generator of <A>G</A> and <Math>Hxg = Hy</Math>. When <Math>g</Math> is an involution, the two directed edges
<Math>(Hx, Hy)</Math> and <Math>(Hy, Hx)</Math> are replaced by a single undirected edge <Math>\{Hx, Hy\}</Math> with label <Math>g</Math>.
<P/>
In the &corefreesub; package, this can achieved by creating graphs as DOT files and using an adaptation of the visualization package developed by M. Delgado et al. <Cite Key="IntPic"/> <Cite Key="Automata"/>, which can be found on Chapter 4.
<P/>
For calculating core-free subgroups of solvable groups, a function <A>CoreFreeConjugacyClassesSubgroupsOfSolvableGroup</A> was written based on the <A>FiniteSubgroupClassesBySeries</A> of <A>polycyclic</A> GAP package <Cite Key="Polycyclic"/>,
under GNU General Public License v2 or above.
<P/>
This package was created using the GAP Package <A>PackageMaker</A> <Cite Key="PackageMaker"/>, with documentation done using <A>AutoDoc</A> <Cite Key="AutoDoc"/>.
<P/>
<Section Label="Chapter_Introduction_Section_Installation">
<Heading>Installation</Heading>
<P/>
To install this package, you can simply copy the folder of &corefreesub; and its contents into your <F>/pkg</F> folder inside your &GAP; installation folder.
This should work for Windows, Ubuntu and MacOS.
If you are using GAP.app on MacOS, the &corefreesub; folder should be copied into your user <F>Library/Preferences/GAP/pkg</F> folder.
<P/>
This package was tested with &GAP; version greater or equal to 4.11.
<P/>
Moreover, this package depends on the <A>polycyclic</A> GAP package <Cite Key="Polycyclic"/> to calculate core-free subgroups of solvable groups.
<P/>
For the graphical output of faithful transitive permutation representation of graphs, <A>graphviz</A> should be installed, as well as
<A>dot2tex</A> for the output of a Tikz-Tex file.
<P/>
</Section>
<Section Label="Chapter_Introduction_Section_Testing_your_installation">
<Heading>Testing your installation</Heading>
<P/>
To test your installation, you can run the function <A>CF_TESTALL()</A>.
This function will run two sets of tests, one dependent on the
documentation of the &corefreesub; package and another with assertions with groups with bigger size.
<P/>
If the test runs with no issue, the output should look something similar to the following:
<Log><![CDATA[
gap> CF_TESTALL();
Running list 1 . . .
gap>
]]></Log>
This tests will also produce two pictures that are supposed to be output and open in the user system.
If the tests run with no error but they do not output any of the graphs, then it may mean the user might not be able to
use this functionality. If so, please report an issue on <URL Text="CoreFreeSub GitHub Issues">https://github.com/CAPiedade/corefreesub/issues</URL>.
<P/>
</Section>
</Chapter>
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