<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroupAsCatOneGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> and returns the Cat-1-group <span class="SimpleMath">C</span> corresponding to the crossed module <span class="SimpleMath">G→ Aut(G)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomotopyGroup</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and an integer n. It returns the <span class="SimpleMath">n</span>th homotopy group of <span class="SimpleMath">C</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomotopyModule</code>( <var class="Arg">C</var>, <var class="Arg">2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and an integer n=2. It returns the second homotopy group of <span class="SimpleMath">C</span> as a G-module (i.e. abelian G-outer group) where G is the fundamental group of C.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuasiIsomorph</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and returns a cat-1-group <span class="SimpleMath">D</span> for which there exists some homomorphism <span class="SimpleMath">C→ D</span> that induces isomorphisms on homotopy groups.</p>
<p>This function was implemented by <strong class="button">Le Van Luyen</strong>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModuleAsCatOneGroup</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span>, an abelian group <span class="SimpleMath">M</span> and a homomorphism <span class="SimpleMath">α: G→ Aut(M)</span>. It returns the Cat-1-group <span class="SimpleMath">C</span> corresponding th the zero crossed module <span class="SimpleMath">0: M→ G</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MooreComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and returns its Moore complex as a G-complex (i.e. as a complex of groups considered as 1-outer groups).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalSubgroupAsCatOneGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> with normal subgroup <span class="SimpleMath">N</span>. It returns the Cat-1-group <span class="SimpleMath">C</span> corresponding th the inclusion crossed module <span class="SimpleMath">N→ G</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XmodToHAP</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> obtained from the Xmod package and returns a cat-1-group <span class="SimpleMath">D</span> for which IsHapCatOneGroup(D) returns true.</p>
<p>It returns "fail" id <span class="SimpleMath">C</span> has not been produced by the Xmod package.</p>
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