  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YThe  [5Xcorefreesub[105X  package  was created to calculate core-free subgroups of a
  group, their indexes, and faithful transitive permutation representations.[133X
  
  [33X[0;0YA core-free subgroup of a group [3XG[103X is a subgroup [3XH[103X such that[133X
  
  
  [24X[33X[0;6Y\cap_{g\in G} H = \{id_G\}.[133X
  
  [124X
  
  [33X[0;0YThese  subgroups  are  important since the action of [3XG[103X on the cosets of [3XH[103X is
  both  transitive  and  faithful.  Hence, this gives us a faithful transitive
  permutation  representation of [3XG[103X with degree [3Xn[103X, where [3Xn[103X is the index of [3XH[103X in
  [3XG[103X.[133X
  
  [33X[0;0YThere  are  many  articles  studying  faithful permutation representation of
  groups, such as [Joh71], [EP88], [Sau14] and [EH16]. However the restriction
  on  transitive  actions  is  more  recent  and  there are fewer studies like
  [FP20],[FP21a],[FP21b] and [FP22].[133X
  
  [33X[0;0YDuring  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.[133X
  
  [33X[0;0YOne  of  the  important  tools  for studying faithful transitive permutation
  representations  is  by using [3Xfaithful transitive permutation representation
  graphs[103X,  which  are [3XSchreier coset graphs[103X. A [3XSchreier coset graph[103X is a graph
  associated  with  a  group  [3XG[103X,  its  generators  and  a subgroup [3XH[103X of [3XG[103X. The
  vertices of the graph are the right cosets of [3XH[103X and there is a directed edge
  [23X(Hx,Hy)[123X  with  label  [23Xg[123X  if [23Xg[123X is a generator of [3XG[103X and [23XHxg = Hy[123X. When [23Xg[123X is an
  involution,  the  two directed edges [23X(Hx, Hy)[123X and [23X(Hy, Hx)[123X are replaced by a
  single undirected edge [23X\{Hx, Hy\}[123X with label [23Xg[123X.[133X
  
  [33X[0;0YIn  the  [5Xcorefreesub[105X  package,  this  can achieved by creating graphs as DOT
  files  and  using an adaptation of the visualization package developed by M.
  Delgado et al. [Del22] [DLM22], which can be found on Chapter 4.[133X
  
  [33X[0;0YFor   calculating   core-free  subgroups  of  solvable  groups,  a  function
  [3XCoreFreeConjugacyClassesSubgroupsOfSolvableGroup[103X  was  written  based on the
  [3XFiniteSubgroupClassesBySeries[103X  of  [3Xpolycyclic[103X GAP package [ENH20], under GNU
  General Public License v2 or above.[133X
  
  [33X[0;0YThis  package  was  created using the GAP Package [3XPackageMaker[103X [Hor19], with
  documentation done using [3XAutoDoc[103X [GH22].[133X
  
  
  [1X1.1 [33X[0;0YInstallation[133X[101X
  
  [33X[0;0YTo  install  this package, you can simply copy the folder of [5Xcorefreesub[105X and
  its contents into your [11X/pkg[111X folder inside your [5XGAP[105X installation folder. This
  should  work  for  Windows,  Ubuntu  and  MacOS. If you are using GAP.app on
  MacOS,   the   [5Xcorefreesub[105X   folder   should   be   copied  into  your  user
  [11XLibrary/Preferences/GAP/pkg[111X folder.[133X
  
  [33X[0;0YThis package was tested with [5XGAP[105X version greater or equal to 4.11.[133X
  
  [33X[0;0YMoreover,  this  package  depends  on  the [3Xpolycyclic[103X GAP package [ENH20] to
  calculate core-free subgroups of solvable groups.[133X
  
  [33X[0;0YFor  the  graphical output of faithful transitive permutation representation
  of  graphs,  [3Xgraphviz[103X should be installed, as well as [3Xdot2tex[103X for the output
  of a Tikz-Tex file.[133X
  
  
  [1X1.2 [33X[0;0YTesting your installation[133X[101X
  
  [33X[0;0YTo  test  your  installation,  you  can  run the function [3XCF_TESTALL()[103X. This
  function  will  run two sets of tests, one dependent on the documentation of
  the  [5Xcorefreesub[105X package and another with assertions with groups with bigger
  size.[133X
  
  [33X[0;0YIf  the test runs with no issue, the output should look something similar to
  the following:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCF_TESTALL();[127X[104X
    [4X[28XRunning list 1 . . .[128X[104X
    [4X[28Xgap>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis 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 CoreFreeSub GitHub
  Issues ([7Xhttps://github.com/CAPiedade/corefreesub/issues[107X).[133X
  
