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tutorial/chap6.html

@@ -400,7 +400,7 @@ <h4>6.8 <span class="Heading">Cocyclic groups: a convenient way of representing
 
 </li>
 </ol>
-<p>As an illustration where the base group is a non-solvable finite group and the fibre is the infinite cyclic group, with base group acting trivially on the fibre, the following commands list up to Yoneda equivalence all central extensions <span class="SimpleMath">Z ↣ E ↠ G</span> for <span class="SimpleMath">G=A_5:C_16</span>. The base group is a non-solvable semi-direct product of order <span class="SimpleMath">960</span> and thus none of the <span class="SimpleMath">16</span> extensions are polycyclic. The commands classify the extensions according to their integral homology in degrees <span class="SimpleMath">≤ 2</span>, showing that there are precisely 5 such equivalence classes of extensions. Thus, there are at least 5 distinct isomorphism types among the <span class="SimpleMath">16</span> extensions. A presentation is constructed for the group corresponding to the sixteenth extension.</p>
+<p>As an illustration where the base group is a non-solvable finite group and the fibre is the infinite cyclic group, with base group acting trivially on the fibre, the following commands list up to Yoneda equivalence all central extensions <span class="SimpleMath">Z ↣ E ↠ G</span> for <span class="SimpleMath">G=A_5:C_16</span>. The base group is a non-solvable semi-direct product of order <span class="SimpleMath">960</span> and thus none of the <span class="SimpleMath">16</span> extensions are polycyclic. The commands classify the extensions according to their integral homology in degrees <span class="SimpleMath">≤ 2</span>, showing that there are precisely 5 such equivalence classes of extensions. Thus, there are at least 5 distinct isomorphism types among the <span class="SimpleMath">16</span> extensions. A presentation is constructed for the group corresponding to the sixteenth extension. The final command lists the orders of the 16 cohomology group elements corresponding to the 16 extensions. The 16th element has order 1, meaning that the sixteenth extension is the direct product <span class="SimpleMath">C_∞ × A_5:C_16</span>.</p>
 
 
 <div class="example"><pre>
@@ -444,6 +444,9 @@ <h4>6.8 <span class="Heading">Cocyclic groups: a convenient way of representing
   w*z*w^-1*z^-1*w^-1*z, z*y^2*(z^-1*y^-1)^2, v^-1*x^-1*v*x, v*y*v^-1*y^-1, 
   v*z*v^-1*z^-1, v*w*v^-1*w^-1 ]
 
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(Elts,Order);</span>
+[ 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 4, 4, 2, 1 ]
+
 </pre></div>
 
 <p><a id="X7C60E2B578074532" name="X7C60E2B578074532"></a></p>

tutorial/chap6.txt

@@ -393,7 +393,10 @@
   homology in degrees ≤ 2, showing that there are precisely 5 such equivalence
   classes of extensions. Thus, there are at least 5 distinct isomorphism types
   among  the  16  extensions.  A  presentation  is  constructed  for the group
-  corresponding to the sixteenth extension.
+  corresponding to the sixteenth extension. The final command lists the orders
+  of  the 16 cohomology group elements corresponding to the 16 extensions. The
+  16th element has order 1, meaning that the sixteenth extension is the direct
+  product C_∞ × A_5:C_16.
 
     Example  
     gap> G:=SmallGroup(960,637);;
@@ -436,6 +439,9 @@
       w*z*w^-1*z^-1*w^-1*z, z*y^2*(z^-1*y^-1)^2, v^-1*x^-1*v*x, v*y*v^-1*y^-1, 
       v*z*v^-1*z^-1, v*w*v^-1*w^-1 ]
     
+    gap> List(Elts,Order);
+    [ 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 4, 4, 2, 1 ]
+    
   
 
 

tutorial/chap6_mj.html

@@ -403,7 +403,7 @@ <h4>6.8 <span class="Heading">Cocyclic groups: a convenient way of representing
 
 </li>
 </ol>
-<p>As an illustration where the base group is a non-solvable finite group and the fibre is the infinite cyclic group, with base group acting trivially on the fibre, the following commands list up to Yoneda equivalence all central extensions <span class="SimpleMath">\(\mathbb Z \rightarrowtail E \twoheadrightarrow G\)</span> for <span class="SimpleMath">\(G=A_5:C_{16}\)</span>. The base group is a non-solvable semi-direct product of order <span class="SimpleMath">\(960\)</span> and thus none of the <span class="SimpleMath">\(16\)</span> extensions are polycyclic. The commands classify the extensions according to their integral homology in degrees <span class="SimpleMath">\( \le 2\)</span>, showing that there are precisely 5 such equivalence classes of extensions. Thus, there are at least 5 distinct isomorphism types among the <span class="SimpleMath">\(16\)</span> extensions. A presentation is constructed for the group corresponding to the sixteenth extension.</p>
+<p>As an illustration where the base group is a non-solvable finite group and the fibre is the infinite cyclic group, with base group acting trivially on the fibre, the following commands list up to Yoneda equivalence all central extensions <span class="SimpleMath">\(\mathbb Z \rightarrowtail E \twoheadrightarrow G\)</span> for <span class="SimpleMath">\(G=A_5:C_{16}\)</span>. The base group is a non-solvable semi-direct product of order <span class="SimpleMath">\(960\)</span> and thus none of the <span class="SimpleMath">\(16\)</span> extensions are polycyclic. The commands classify the extensions according to their integral homology in degrees <span class="SimpleMath">\( \le 2\)</span>, showing that there are precisely 5 such equivalence classes of extensions. Thus, there are at least 5 distinct isomorphism types among the <span class="SimpleMath">\(16\)</span> extensions. A presentation is constructed for the group corresponding to the sixteenth extension. The final command lists the orders of the 16 cohomology group elements corresponding to the 16 extensions. The 16th element has order 1, meaning that the sixteenth extension is the direct product <span class="SimpleMath">\(C_\infty\ \times\ A_5:C_{16}\)</span>.</p>
 
 
 <div class="example"><pre>
@@ -447,6 +447,9 @@ <h4>6.8 <span class="Heading">Cocyclic groups: a convenient way of representing
   w*z*w^-1*z^-1*w^-1*z, z*y^2*(z^-1*y^-1)^2, v^-1*x^-1*v*x, v*y*v^-1*y^-1, 
   v*z*v^-1*z^-1, v*w*v^-1*w^-1 ]
 
+<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">List(Elts,Order);</span>
+[ 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 4, 4, 2, 1 ]
+
 </pre></div>
 
 <p><a id="X7C60E2B578074532" name="X7C60E2B578074532"></a></p>