<?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 18: Cat-1-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="chap17.html">Previous Chapter</a> <a href="chap19.html">Next Chapter</a> </div> <p><a id="X7B54B8CA841C517B" name="X7B54B8CA841C517B"></a></p> <div class="ChapSects"><a href="chap18.html#X7B54B8CA841C517B">18. <span class="Heading"> Cat-1-groups</span></a> </div> <h3>18. <span class="Heading"> Cat-1-groups</span></h3> <div class="pcenter"><table cellspacing="10" class="GAPDocTable"> <tr> <td class="tdleft"><code class="code"> AutomorphismGroupAsCatOneGroup(G)</code></p> <p>Inputs a group G and returns the Cat-1-group C corresponding th the crossed module G-> Aut(G).</td> </tr> <tr> <td class="tdleft"><code class="code">HomotopyGroup(C,n)</code></p> <p>Inputs a cat-1-group C and an integer n. It returns the nth homotopy group of C.</td> </tr> <tr> <td class="tdleft"><code class="code">HomotopyModule(C,2)</code></p> <p>Inputs a cat-1-group C and an integer n=2. It returns the second homotopy group of C as a G-module (i.e. abelian G-outer group) where G is the fundamental group of C.</td> </tr> <tr> <td class="tdleft"><code class="code"> ModuleAsCatOneGroup(G,alpha,M)</code></p> <p>Inputs a group G, an abelian group M and a homomorphism alpha: G-> Aut(M). It returns the Cat-1-group C corresponding th the zero crossed module 0: M-> G.</td> </tr> <tr> <td class="tdleft"><code class="code"> MooreComplex(C)</code></p> <p>Inputs a cat-1-group C and returns its Moore complex [M_1 -> M_0] as a list whose single entry is a homomorphism of groups.</td> </tr> <tr> <td class="tdleft"><code class="code"> NormalSubgroupAsCatOneGroup(G,N)</code></p> <p>Inputs a group G with normal subgroup N. It returns the Cat-1-group C corresponding th the inclusion crossed module N-> G.</td> </tr> </table><br /><p> </p><br /> </div> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap17.html">Previous Chapter</a> <a href="chap19.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>