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<div class="ChapSects"><a href="chap23.html#X7D02CE0A83211FB7">23 <span class="Heading"> G-Outer Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23.html#X7CFDEEC07F15CF82">23.1 <span class="Heading"> </span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap23.html#X842035BD7E0B81EF">23.1-1 GOuterGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap23.html#X7F681DB67F556FDF">23.1-2 GOuterGroupHomomorphismNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap23.html#X7B4CE3397CAED0EC">23.1-3 GOuterHomomorphismTester</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap23.html#X847ABE6F781C7FE8">23.1-4 Centre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap23.html#X7F5C49A38455A64C">23.1-5 DirectProductGog</a></span>
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<h3>23 <span class="Heading"> G-Outer Groups</span></h3>
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<h4>23.1 <span class="Heading"> </span></h4>
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<h5>23.1-1 GOuterGroup</h5>
<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>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoefficientSequence.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">4</a></span> </p>
<p><a id="X7F681DB67F556FDF" name="X7F681DB67F556FDF"></a></p>
<h5>23.1-2 GOuterGroupHomomorphismNC</h5>
<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>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7B4CE3397CAED0EC" name="X7B4CE3397CAED0EC"></a></p>
<h5>23.1-3 GOuterHomomorphismTester</h5>
<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>
<p><strong class="button">Examples:</strong></p>
<p><a id="X847ABE6F781C7FE8" name="X847ABE6F781C7FE8"></a></p>
<h5>23.1-4 Centre</h5>
<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>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutParallel.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">6</a></span> </p>
<p><a id="X7F5C49A38455A64C" name="X7F5C49A38455A64C"></a></p>
<h5>23.1-5 DirectProductGog</h5>
<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>
<p><strong class="button">Examples:</strong></p>
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