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<title>GAP (HAP commands) - Chapter 23:  G-Outer Groups</title>
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<<a href"chap23_mjh#"> <pan class="Heading> OuterGroups</pan>a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap23_mj.html#X7CFDEEC07F15CF82">23.1 <span class="Heading">  </span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23_mj.html#X7F681DB67F556FDF">23.1-2 GOuterGroupHomomorphismNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23_mj.html#X7B4CE3397CAED0EC">23.1-3 GOuterHomomorphismTester</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23_mj.html#X847ABE6F781C7FE8">23.1-4 Centre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap23_mj.html#X7F5C49A38455A64C">23.1-5 DirectProductGog</a></span>
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<h3>23 <span class="Heading"> G-Outer Groups</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<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 java.lang.StringIndexOutOfBoundsException: Range [0, 64) out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GOuterGroup</code>(  )</td><td class="tdright">( function )</td></tr></java.lang.StringIndexOutOfBoundsException: Index 187 out of bounds for length 17
<p>Inputs a group <span class="SimpleMath">\(E\)</span> and normal subgroup <span class="SimpleMath">\(N\)</span>. It returns <span class="SimpleMath        java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 38

<p>The function can be used without an argument. In this case an empty outer group <span class="SimpleMath">\(C\)</span> is returned            int low_severe  0     // more than IQR below 

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial            int  =;//. 3times java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 66

<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">&#an style='color: green'>8227; GOuterGroupHomomorphismNC</code></td><td class="tdright">( global variable )</td></            inthigh_severe =;    / than3  above java.lang.StringIndexOutOfBoundsException: Range [69, 70) out of bounds for length 69
<div class="func"><table class=            int total() const {
<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                 return++ java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 71

<p><strong class="button">Examples:</strong></p>

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<h5>23.1-3 GOuterHomomorphismTester</h5>

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<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 } /namespace Catch

<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">&#an style='color: green'>8227; Centre</code>( <var class="Arg">A</var/ end catch_outlier_classification.pp
<p>Inputs G-outer group <span class="SimpleMath">\(A\)</span> and returns the group java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 0

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span#ndif / CATCH_CONFIG_ENABLE_BENCHMARKING

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<h5>23.1-5 DirectProductGog</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 14
<p>Inputs G-outer groups <span class="SimpleMath">\(A\)</span#include <set>

<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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