<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GOuterGroup</code>( <var class="Arg">E</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GOuterGroup</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(E\)</span> and normal subgroup <span class="SimpleMath">\(N\)</span>. It returns <span class="SimpleMath">\(N\)</span> as a <span class="SimpleMath">\(G\)</span>-outer group where <span class="SimpleMath">\(G=E/N\)</span>.</p>
<p>The function can be used without an argument. In this case an empty outer group <span class="SimpleMath">\(C\)</span> is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GOuterGroupHomomorphismNC</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GOuterGroupHomomorphismNC</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs G-outer groups <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> with common acting group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). It returns the corresponding G-outer homomorphism PHI:A--> B. No check is made to verify that phi is actually a group homomorphism which preserves the G-action.</p>
<p>The function can be used without an argument. In this case an empty outer group homomorphism <span class="SimpleMath">\(PHI\)</span> is returned. The components must then be set.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GOuterHomomorphismTester</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">phi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs G-outer groups <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> with common acting group, and a group homomorphism phi:ActedGroup(A) --> ActedGroup(B). It tests whether phi is a group homomorphism which preserves the G-action.</p>
<p>The function can be used without an argument. In this case an empty outer group homomorphism <span class="SimpleMath">\(PHI\)</span> is returned. The components must then be set.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Centre</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs G-outer group <span class="SimpleMath">\(A\)</span> and returns the group theoretic centre of ActedGroup(A) as a G-outer group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectProductGog</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectProductGog</code>( <var class="Arg">Lst</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs G-outer groups <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> with common acting group, and returns their group-theoretic direct product as a G-outer group. The outer action on the direct product is the diagonal one.</p>
<p>The function also applies to a list Lst of G-outer groups with common acting group.</p>
<p>For a direct product D constructed using this function, the embeddings and projections can be obtained (as G-outer group homomorphisms) using the functions Embedding(D,i) and Projection(D,i).</p>
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