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<p><a id="X78D8803A84DB6CE8" name="X78D8803A84DB6CE8"></a></p>
<div class="ChapSects"><a href="chap2.html#X78D8803A84DB6CE8">2 <span class="Heading">Obtaining Core-Free Subgroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X79EF503F837B7702">2.1 <span class="Heading">Core-Free Subgroups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A9042C27F0BA85B">2.1-1 IsCoreFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D0391668782765C">2.1-2 CoreFreeConjugacyClassesSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X821F9BD687383C1F">2.1-3 AllCoreFreeSubgroups</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X847B222E84342E88">2.2 <span class="Heading">Degrees of Core-Free subgroups</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X844F04457C6EDC67">2.2-1 CoreFreeDegrees</a></span>
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<h3>2 <span class="Heading">Obtaining Core-Free Subgroups</span></h3>

<p><a id="X79EF503F837B7702" name="X79EF503F837B7702"></a></p>

<h4>2.1 <span class="Heading">Core-Free Subgroups</span></h4>

<p>A core-free subgroup is a subgroup in which its (normal) core is trivial.</p>

<p><a id="X7A9042C27F0BA85B" name="X7A9042C27F0BA85B"></a></p>

<h5>2.1-1 IsCoreFree</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCoreFree</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a boolean</p>

<p>Given a group <var class="Arg">G</var> and one of its subgroups <var class="Arg">H</var>, it returns whether <var class="Arg">H</var> is core-free in <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SymmetricGroup(4);; H := Subgroup(G, [(1,3)(2,4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Core(G,H);</span>
Group(())
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCoreFree(G,H);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Subgroup(G, [(1,4)(2,3), (1,3)(2,4)]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCoreFree(G,H);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Core(G,H);# H is a normal subgroup of G, hence it does not have a trivial core</span>
Group([ (1,4)(2,3), (1,3)(2,4) ])
</pre></div>

<p><a id="X7D0391668782765C" name="X7D0391668782765C"></a></p>

<h5>2.1-2 CoreFreeConjugacyClassesSubgroups</h5>

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

<p>Returns a list of all conjugacy classes of core-free subgroups of <var class="Arg">G</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SymmetricGroup(4);; dh := DihedralGroup(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">cfccs := CoreFreeConjugacyClassesSubgroups(G);; Size(cfccs);</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">cfccs_dh := CoreFreeConjugacyClassesSubgroups(dh);; Size(cfccs_dh);</span>
2
</pre></div>

<p><a id="X821F9BD687383C1F" name="X821F9BD687383C1F"></a></p>

<h5>2.1-3 AllCoreFreeSubgroups</h5>

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

<p>Returns a list of all core-free subgroups of <var class="Arg">G</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SymmetricGroup(4);; dh := DihedralGroup(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">acfs := AllCoreFreeSubgroups(G);; Size(acfs);</span>
24
<span class="GAPprompt">gap></span> <span class="GAPinput">acfs_dh := AllCoreFreeSubgroups(dh);; Size(acfs_dh);</span>
6
</pre></div>

<p><a id="X847B222E84342E88" name="X847B222E84342E88"></a></p>

<h4>2.2 <span class="Heading">Degrees of Core-Free subgroups</span></h4>

<p><a id="X844F04457C6EDC67" name="X844F04457C6EDC67"></a></p>

<h5>2.2-1 CoreFreeDegrees</h5>

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

<p>Returns a list of all possible degrees of faithful transitive permutation representations of <var class="Arg">G</var>. The degrees of a faithful transitive permutation representation of <var class="Arg">G</var> are the index of its core-free subgroups.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SymmetricGroup(4);; dh := DihedralGroup(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CoreFreeDegrees(G);</span>
[ 4, 6, 8, 12, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CoreFreeDegrees(dh);</span>
[ 5, 10 ]
</pre></div>


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