From 858146b6f0e80d32ffff426d12a1fb298f06b4c0 Mon Sep 17 00:00:00 2001 From: grahamknockillaree Date: Mon, 30 Dec 2024 01:13:23 +0000 Subject: [PATCH] Add files via upload --- tutorial/chap6.html | 5 ++++- tutorial/chap6.txt | 8 +++++++- tutorial/chap6_mj.html | 5 ++++- 3 files changed, 15 insertions(+), 3 deletions(-) diff --git a/tutorial/chap6.html b/tutorial/chap6.html index e902b648..e1c166cb 100644 --- a/tutorial/chap6.html +++ b/tutorial/chap6.html @@ -400,7 +400,7 @@

6.8 Cocyclic groups: a convenient way of representing -

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 Z ↣ E ↠ G for G=A_5:C_16. The base group is a non-solvable semi-direct product of order 960 and thus none of the 16 extensions are polycyclic. The commands classify the extensions according to their integral 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.

+

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 Z ↣ E ↠ G for G=A_5:C_16. The base group is a non-solvable semi-direct product of order 960 and thus none of the 16 extensions are polycyclic. The commands classify the extensions according to their integral 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. 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.

@@ -444,6 +444,9 @@ 
6.8 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 ]
 
+gap> List(Elts,Order);
+[ 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 4, 4, 2, 1 ]
+

diff --git a/tutorial/chap6.txt b/tutorial/chap6.txt index 27dc2995..a6586617 100644 --- a/tutorial/chap6.txt +++ b/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 ] +   diff --git a/tutorial/chap6_mj.html b/tutorial/chap6_mj.html index 79c31c25..2a7fac4c 100644 --- a/tutorial/chap6_mj.html +++ b/tutorial/chap6_mj.html @@ -403,7 +403,7 @@

6.8 Cocyclic groups: a convenient way of representing -

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 \(\mathbb Z \rightarrowtail E \twoheadrightarrow G\) for \(G=A_5:C_{16}\). The base group is a non-solvable semi-direct product of order \(960\) and thus none of the \(16\) extensions are polycyclic. The commands classify the extensions according to their integral homology in degrees \( \le 2\), 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.

+

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 \(\mathbb Z \rightarrowtail E \twoheadrightarrow G\) for \(G=A_5:C_{16}\). The base group is a non-solvable semi-direct product of order \(960\) and thus none of the \(16\) extensions are polycyclic. The commands classify the extensions according to their integral homology in degrees \( \le 2\), 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. 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 \(C_\infty\ \times\ A_5:C_{16}\).

@@ -447,6 +447,9 @@ 
6.8 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 ]
 
+gap> List(Elts,Order);
+[ 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 4, 4, 2, 1 ]
+