<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (HAP) - Chapter 17: G-Outer Groups</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body><a href="../www/index.html"><small>HAP home</small></a> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap16.html">Previous Chapter</a> <a href="chap18.html">Next Chapter</a> </div> <p><a id="X7D02CE0A83211FB7" name="X7D02CE0A83211FB7"></a></p> <div class="ChapSects"><a href="chap17.html#X7D02CE0A83211FB7">17. <span class="Heading"> G-Outer Groups</span></a> </div> <h3>17. <span class="Heading"> G-Outer Groups</span></h3> <div class="pcenter"><table cellspacing="10" class="GAPDocTable"> <tr> <td class="tdleft"><code class="code">GOuterGroup(E,N)</code> <br /> <code class="code">GOuterGroup()</code></p> <p>Inputs a group E and normal subgroup N. It returns N as a G-outer group where G=E/N.</p> <p>The function can be used without an argument. In this case an empty outer group C is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).</td> </tr> <tr> <td class="tdleft"><code class="code">GOuterGroupHomomorphismNC(A,B,phi)</code> <br /> <code class="code">GOuterGroupHomomorphismNC()</code></p> <p>Inputs G-outer groups A and B 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 PHI is returned. The components must then be set.</td> </tr> <tr> <td class="tdleft"><code class="code">GOuterHomomorphismTester(A,B,phi)</code></p> <p>Inputs G-outer groups A and B 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 PHI is returned. The components must then be set.</td> </tr> <tr> <td class="tdleft"><code class="code">Centre(A)</code></p> <p>Inputs G-outer group A and returns the group theoretic centre of ActedGroup(A) as a G-outer group.</td> </tr> <tr> <td class="tdleft"><code class="code">DirectProductGog(A,B)</code> <br /> <code class="code">DirectProductGog(Lst)</code></p> <p>Inputs G-outer groups A and B 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).</td> </tr> </table><br /><p> </p><br /> </div> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap16.html">Previous Chapter</a> <a href="chap18.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>