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<title>GAP (corefreesub) - Chapter 3: Faithful Transitive Permutation Representations</title>
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<p><a id="X857DD6BF86E302D0" name="X857DD6BF86E302D0"></a></p>
<div class="ChapSects"><a href="chap3.html#X857DD6BF86E302D0">3 <span class="Heading">Faithful Transitive Permutation Representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X839E0E8B87479AB5">3.1 <span class="Heading">Obtaining Faithful Transitive Permutation Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80D610507B1029C9">3.1-1 FaithfulTransitivePermutationRepresentations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X836069DB849BD28D">3.2 <span class="Heading">Faithful Transitive Permutation Representation of Minimal Degree</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8746DBA2866336CC">3.2-1 MinimalFaithfulTransitivePermutationRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B0EBFFA86B38A0F">3.2-2 MinimalFaithfulTransitivePermutationDegree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X845EED7E826D411B">3.3 <span class="Heading">Faithful Transitive Permutation Representation of given Degree</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7EC3C43C80AAD2CB">3.3-1 FaithfulTransitivePermutationRepresentationsOfDegree</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">Faithful Transitive Permutation Representations</span></h3>

<p>The action of a group G on the coset space of a subgroup gives us a transitive permutation representation of the group. Whenever the subgroup is core-free, we have that the action of G on the coset space of the subgroup will be faithful. Moreover, the stabilizer of a point on a faithful transitive permutation representation of G will always be a core-free subgroup.</p>

<p><a id="X839E0E8B87479AB5" name="X839E0E8B87479AB5"></a></p>

<h4>3.1 <span class="Heading">Obtaining Faithful Transitive Permutation Representations</span></h4>

<p><a id="X80D610507B1029C9" name="X80D610507B1029C9"></a></p>

<h5>3.1-1 FaithfulTransitivePermutationRepresentations</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FaithfulTransitivePermutationRepresentations</code>( <var class="Arg">G</var>[, <var class="Arg">all_ftpr</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list</p>

<p>For a finite group <var class="Arg">G</var>, <var class="Arg">FaithfulTransitivePermutationRepresentations</var> returns a list of a faithful transitive permutation representation of <var class="Arg">G</var> for each degree. If <var class="Arg">all_ftpr</var> is true, then it will return a list of all faithful transitive permutation representations, up to conjugacy equivalence.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sp := SymplecticGroup(4,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CoreFreeDegrees(sp);</span>
[ 6, 10, 12, 15, 20, 30, 36, 40, 45, 60, 72, 80, 90, 120, 144, 180, 240, 360, 
  720 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ftprs := FaithfulTransitivePermutationRepresentations(sp);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(ftprs);</span>
19
<span class="GAPprompt">gap></span> <span class="GAPinput">all_ftprs := FaithfulTransitivePermutationRepresentations(sp,true);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(all_ftprs);</span>
54
</pre></div>

<p><a id="X836069DB849BD28D" name="X836069DB849BD28D"></a></p>

<h4>3.2 <span class="Heading">Faithful Transitive Permutation Representation of Minimal Degree</span></h4>

<p>To complement the already existing functions in GAP <var class="Arg">MinimalFaithfulPermutationDegree</var> and <var class="Arg">MinimalFaithfulPermutationRepresentation</var>, the following functions to retrieve the <var class="Arg">MinimalFaithfulTransitivePermutationRepresentation</var> and <var class="Arg">MinimalFaithfulTransitivePermutationDegree</var>.</p>

<p><a id="X8746DBA2866336CC" name="X8746DBA2866336CC"></a></p>

<h5>3.2-1 MinimalFaithfulTransitivePermutationRepresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalFaithfulTransitivePermutationRepresentation</code>( <var class="Arg">G</var>[, <var class="Arg">all_minimal_ftpr</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an isomorphism (or a list of isomorphisms)</p>

<p>For a finite group <var class="Arg">G</var>, <var class="Arg">MinimalFaithfulTransitivePermutationRepresentation</var> returns an isomorphism of <var class="Arg">G</var> into the symmetric group of minimal degree acting transitively on its domain. If <var class="Arg">all_minimal_ftpr</var> is set as <var class="Arg">true</var>, then it returns a list of all isomorphisms <var class="Arg">G</var> into the symmetric group of minimal degree.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sp := SymplecticGroup(4,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">min_ftpr := MinimalFaithfulTransitivePermutationRepresentation(sp);</span>
CompositionMapping( <action epimorphism>, <action isomorphism> )
<span class="GAPprompt">gap></span> <span class="GAPinput">min_ftprs := MinimalFaithfulTransitivePermutationRepresentation(sp,true);</span>
[ CompositionMapping( <action epimorphism>, <action isomorphism> ), 
  CompositionMapping( <action epimorphism>, <action isomorphism> ) ]
</pre></div>

<p><a id="X7B0EBFFA86B38A0F" name="X7B0EBFFA86B38A0F"></a></p>

<h5>3.2-2 MinimalFaithfulTransitivePermutationDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalFaithfulTransitivePermutationDegree</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an integer</p>

<p>For a finite group <var class="Arg">G</var>, <var class="Arg">MinimalFaithfulTransitivePermutationDegree</var> returns the least positive integer n such that <var class="Arg">G</var> is isomorphic to a subgroup of the symmetric group of degree n acting transitively on its domain.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sp := SymplecticGroup(4,2);; g:=SimpleGroup("PSL",3,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalFaithfulTransitivePermutationDegree(sp);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalFaithfulTransitivePermutationDegree(g);</span>
31
</pre></div>

<p><a id="X845EED7E826D411B" name="X845EED7E826D411B"></a></p>

<h4>3.3 <span class="Heading">Faithful Transitive Permutation Representation of given Degree</span></h4>

<p>To obtain a faithful transitive permutation Representation of a specific degree, the following function <var class="Arg">FaithfulTransitivePermutationRepresentationsOfDegree</var> can be used.</p>

<p><a id="X7EC3C43C80AAD2CB" name="X7EC3C43C80AAD2CB"></a></p>

<h5>3.3-1 FaithfulTransitivePermutationRepresentationsOfDegree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FaithfulTransitivePermutationRepresentationsOfDegree</code>( <var class="Arg">G</var>, <var class="Arg">d</var>[, <var class="Arg">all_ftpr_of_given_degree</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an isomorphism (or a list of isomorphisms)</p>

<p>For a finite group <var class="Arg">G</var>, <var class="Arg">FaithfulTransitivePermutationRepresentationsOfDegree</var> returns one isomorphism of <var class="Arg">G</var> into the symmetric group of degree <var class="Arg">d</var> acting transitively on its domain. If <var class="Arg">all_ftpr_of_given_degree</var> is set as <var class="Arg">true</var>, then it returns a list of all isomorphisms <var class="Arg">G</var> into the symmetric group of degree <var class="Arg">d</var>, up to conjugacy equivalence.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sp := SymplecticGroup(4,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FaithfulTransitivePermutationRepresentationsOfDegree(sp,10);</span>
CompositionMapping( <action epimorphism>, <action isomorphism> )
<span class="GAPprompt">gap></span> <span class="GAPinput">FaithfulTransitivePermutationRepresentationsOfDegree(sp,20, true);</span>
[ CompositionMapping( <action epimorphism>, <action isomorphism> ), 
  CompositionMapping( <action epimorphism>, <action isomorphism> ), 
  CompositionMapping( <action epimorphism>, <action isomorphism> ) ]
</pre></div>


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