Quelle aboutRosenbergerMonster.html Sprache: HTML

products/sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/aboutRosenbergerMonster.html

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
 <meta http-equiv="content-type"
content="text/html; charset=ISO-8859-1">
 <title>AboutHap</title>
</head>
<body
style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
alink="#000066" link="#000066" vlink="#000066">
 
<table
style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
border="0" cellpadding="20" cellspacing="10">
 <tbody>
 <tr align="center">
 <th style="vertical-align: top;">
 <table style="width: 100%; text-align: left;" cellpadding="2"
cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
href="aboutSpaceGroup.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: The
Rosenberger monster - an example of how to piece together information 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCocycles.html">next</a> 
 </td>
 </tr>
 </tbody>
 </table>
 <big></big> 
 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">A
 generalized triangle group
is a group given by a presentation 
 <div style="text-align: center;"> 
< a, b | ap = bq = Rm = 1 > 
 
 </div>
where p, q , m are integers greater than 1 and R is a reduced word in
the free monoid on a and b. The finite generalized triangle groups were
classified in 
 <ul>
 <li>[J. Howie, V.Metaftsis & R.M.Thomas, Triangle groups
and their generalizations, Groups
Korea '94, pages 135-147, (de Gruyter 1995)],
 <li>[L. Levai, G. Rosenberger and B. Souvignier, All finite
generalized triangle groups, Transactions
American Mathematical Society 347 (9) (1995), 3625-3627].</li>
 </ul>
The largest finite generalized triangle group is 
 
 <div style="text-align: center;">G = < a, b
| a2, b3, (abababab2ab2abab2ab2)2
> 
 </div>
 
and has order |G| = 220 34 5 . It has been named
the Rosenberger monster. The
following GAP commands, which were shown to me by R.F. Morse, create a
solvable subgroup K in G of index 5. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
F := FreeGroup("a","b");; a:=F.1;; b:=F.2;; 
gap> R := [a^2, b^3, (a*b*a*b*a*b*a*b^2*a*b^2*a*b*a*b^2*a*b^2)^2];; 
gap> G := F/R;; 
 
gap> a:= G.1;; b:=G.2;; 
gap> K:=Subgroup(G, [a, (b^a)^b, (b^a)^(b^2)]);; 
gap> Index(G,K); 
5 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
Schur Multiplier H2(G,Z) of the Rosenberger monster is a
quotient of H2(K,Z) since K contains the Sylow 2-subgroup
and Sylow 3-subgroup of G, and the Sylow 5-subgroup is cyclic and thus
has trivial Schur multiplier. The following commands first create a
pc-group Kpc isomorphism to K, then construct three terms of a
resolution for K, and finally show that H1(K,Z)=H2(K,Z)
= Z3+Z6 . 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
K_iso_Kfp := IsomorphismFpGroup(K);; 
gap> Kfp:=Image(K_iso_Kfp);; 
gap> Kfp_iso_Kpc:= EpimorphismSolvableQuotient(Kfp,2^20*3^4);; 
gap> Kpc:=Image(Kfp_iso_Kpc);; 
 
gap> D:=DerivedSeries(Kpc)[3];; 
gap> NatHom:=NaturalHomomorphismByNormalSubgroup(Kpc,D);; 
gap> Q:=Image(NatHom);; 
 
gap> N:=NormalSubgroups(Q)[12];; 
gap> RQ:=ResolutionNormalSeries([Q,N,TrivialSubgroup(Q)],3);; 
 
gap> RD:=ResolutionNilpotentGroup(D,3);; 
gap> RKpc:=ResolutionExtension(NatHom,RD,RQ);; 
 
gap> TRKpc:=TensorWithIntegers(RKpc);; 
gap> Homology(TRKpc,1); 
[ 3, 6 ] 
 
gap> Homology(TRKpc,2); 
[ 3, 6 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands construct the group D=[[G,G],[G,G]] and show that H2(G/D,Z)=Z6
and D=[G,D] .</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
D:=DerivedSubgroup(DerivedSubgroup(G));; 
gap> GroupHomology(G/D,2); 
[ 2, 3 ] 
gap> Index(G,D)=Index(G,CommutatorSubgroup(G,D)); 
true 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
five term exact sequence in integral homology (see <a
href="aboutSchurMultiplier.html">this page</a> for details) arising
from the normal subgroup D in G yields a surjection H2(G,Z)
→ H2(G/D,Z) . We conclude that the Schur multiplier of the
Rosenberger monster is either 
 
 <div style="text-align: center;">H2(G,Z) = Z3+Z6
 or 
H2(G,Z) = Z6 .. 
 <div style="text-align: left;"> 
We can complete the calculation of H2(G,Z) using the
following basic result. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">For
any finite group G with presentation G = < x | r > the minimum number
of generator of the Schur multiplier dH2(G,Z) satisfies 
 
 <div style="text-align: center;">dH2(G,Z) < |r| - |x| + 1 . 
 </div>
 </td>
 </tr>
 </tbody>
 </table>
 
This result follows from the fact that H*(G,Z) is finite and
is the homology of a chain complex of free abelian groups with 1
generator in dimension 0, |x| generators in dimension 1 and |r| generators in dimension
2. We now have the following. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="0">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);">Proposition 
 
 The Rosenberger monster has Schur multiplier H2(G,Z)
= Z6 . 
 </td>
 </tr>
 </tbody>
 </table>
 
 </div>
 </div>
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">
 <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutSpaceGroup.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
 </td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCocycles.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
</body>
</html>

Messung V0.5

¤ Dauer der Verarbeitung: 0.11 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.