Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma

Benutzer


products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/ (GAP Algebra Version 4.15.1^©) Datei vom 19.6.2025 mit Größe 22 kB

Quelle aboutIntro.html Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/aboutIntro.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="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top; color: rgb(0, 0, 102);"><a
href="aboutContents.html">Table
Of 
Contents</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; font-weight: bold; color: rgb(0, 0, 102);"><big>About
HAP: Overview 
 </big></td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><a
href="aboutDefinitions.html">Next</a> 
 </td>
 </tr>
 </tbody>
 </table>
 
 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">HAP
can be used to make basic calculations in the cohomology of finite and
infinite groups.
For example, to calculate the integral homology Hn(D201,Z)
of the dihedral group of order 402 in dimension n=99 we could perform
the following commands. 
 </td>
 </tr>
 <tr>
 <td
style="width: 30%; background-color: rgb(255, 255, 204); vertical-align: top;">gap>
F:=FreeGroup(2);; x:=F.1;; y:=F.2;; 
 
gap>
G:=F/[x^2,y^201,(x*y)^2];; G:=Image(IsomorphismPermGroup(G));; 
 
gap> GroupHomology(G,99); 
[ 2, 3, 67 ] 
 
gap> time; 
4845 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
HAP command GroupHomology(G,n)
returns the abelian group
invariants of the n-dimensional homology of the group G with
coefficients in the integers Z with trivial G-action. We see that H99(D201,Z)
= Z402, (Timings are in milliseconds, and most are measured
on a
1.4GHz laptop with 256MB memory.) 
 
The above example has two features that dramatically help the
computations.
Firstly, D201 is a relatively small group. Secondly, D201
has periodic homology with period 4 (meaning that Hn(D201,Z)
= Hn+4(D201,Z) for n>0)
and so the homology groups themselves are small. 
 
Typically, the homology of larger non-periodic groups can only
be computed in low dimensions. The following commands show that: 
 <ul>
 <li>the
alternating group A7 (of order 2520) has H10(A7,Z)
= Z6+(Z3)2 ,</li>
 </ul>
 <ul>
 <li>the special linear group SL3(Z3) (of
order 5616) has H8(SL3(Z3),Z) = Z6
,</li>
 </ul>
 <ul>
 <li>the Coxeter group B5 (of order 3840),
represented by a Coxeter diagram on 5 vertices, has H4(B5,Z)
= (Z2)12
.</li>
 <li>the group K=Ker( SL2(Z53)
→ SL2(Z5) ) (of order 15625) has H3(K,Z)
= (Z5)6+Z125.
(This reproduces a calculation of W.Browder and J.Pakianathan which was
used to produce a <a href="aboutExtensions.html#Adem">counter-example</a>
to a conjecture of A. Adem.) 
 </li>
 <li>the abelian group G=C2×C4×C6×C8×C10
 ×C12 (of order 46080) has H6(G,Z)
= (Z2)280+(Z4)12+(Z12)3
. 
 </li>
 <li>the Mathieu simple group M23 (of order 10200960)
has H2(M23,Z) = H3(M23,Z) =
H4(M23,Z) =
0; this reproduces J. Milgram's <a target="_Loday"
href="aboutFunctorial.html#Milgram">counter-example</a> to a
conjecture of
J.-L. Loday. Furthermore, we get the new result that H5(M23,Z)
= Z7. 
 </li>
 <li>The Mathieu simple group M24 (of order
244823040)
has H3(M24,Z) = Z12 and H4(M24,Z)
= 0. </li>
 </ul>
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
GroupHomology(AlternatingGroup(7),10);time; 
[ 2, 3, 3, 3 ] 
1756 
 
gap> S:=Image(IsomorphismPermGroup(SL(3,3)));; 
gap> GroupHomology(S,8);time; 
[ 2, 3 ] 
6340 
 
gap> B5:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,4]]];; 
gap> GroupHomology(["Coxeter",D],4);time; 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 
56240 
 
gap> K:=MaximalSubgroups(SylowSubgroup(SL(2,Integers mod
5^3),5))[2]; 
gap> K:=Image(IsomorphismPcGroup(K)); 
gap> GroupHomology(K,3);time; 
[ 5, 5, 5, 5, 5, 5, 125 ] 
3254 
 
gap> G:=AbelianGroup([2,4,6,8,10,12]);; 
gap> GroupHomology(G,6);time; 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 12, 12 ] 
23265 
 
gap> GroupHomology(MathieuGroup(23),2);time; 
[ ] 
9395 
gap> GroupHomology(MathieuGroup(23),3);time; 
[ ] 
157961 
gap> GroupHomology(MathieuGroup(23),4);time; 
[ ] 
276853 
gap> GroupHomology(MathieuGroup(23),5);time; 
[ 7 ] 
20639802 
 
gap> GroupHomology(MathieuGroup(24),3);time; 
[ 4, 3 ] 
3205565 
 
gap> GroupHomology(MathieuGroup(24),4); 
[ ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
command GroupHomology()
returns the mod p homology when an optional third argument is set equal
to a prime p. The following shows that the Sylow 2-subgroup P of the
Mathieu simple group M24 has 6-dimensional mod 2 homology H6(P,Z2)=(Z2)143
 . (The group P has order 1024 and the computation took over
two hours to complete.) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
GroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
mod 2 cohomology ring H*(G,Z2) can be calculated
for smallish 2-groups G using the HAPprime extension package (which
uses the Singular system for commutative algebra). For instance, the
following commands compute a presentation and Poincare series for this
ring when G is the Sylow 2-subgroup of the Mathieu group M12 .
The commands use the Lyndon-Hochschild-Serre spectral sequence and
Groebner bases to verify that the computations are correct. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SylowSubgroup(MathieuGroup(12),2);; 
 
gap> Mod2CohomologyRingPresentation(G); 
Graded algebra GF(2)[ x_1, x_2, x_3, x_4, x_5, x_6, x_7 ] / 
[ x_2*x_3, x_1*x_3, x_3*x_4, x_1*x_2^2+x_2^3+x_2*x_5,
x_1*x_2*x_5+x_2*x_6, 
 x_1^2*x_4+x_2^2*x_4+x_2^2*x_5+x_1*x_6+x_4^2+x_4*x_5, 

x_1^3*x_2+x_2^4+x_1*x_2*x_4+x_2^2*x_4+x_2^2*x_5+x_1*x_6+x_2*x_6+x_4*x_5, 
 x_1*x_4*x_5+x_4*x_6, x_2^3*x_5+x_2^2*x_6+x_2*x_5^2, 
 x_1*x_5*x_6+x_3^2*x_7+x_3*x_5*x_6+x_6^2, 

x_2^2*x_4*x_5+x_2^2*x_5^2+x_1*x_4*x_6+x_3^2*x_7+x_3*x_5*x_6+x_4^2*x_5+x_4*x_\ 
5^2+x_6^2, x_2^2*x_4^2+x_2^2*x_5^2+x_2*x_4*x_6+x_2*x_5*x_6, 

x_1^2*x_2*x_6+x_2^3*x_6+x_2^2*x_5^2+x_1*x_4*x_6+x_2*x_4*x_6+x_2*x_5*x_6+x_4^\ 
2*x_5 ] with indeterminate degrees [ 1, 1, 1, 2, 2, 3, 4 ] 
gap> time; 
19685 
 
gap> G:=SylowSubgroup(MathieuGroup(12),2);; 
gap> PoincareSeriesLHS(G); 
(1)/(-x_1^3+3*x_1^2-3*x_1+1) 
gap> time; 
11757 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
homology of certain infinite groups can also be calculated. The
following commands show that 
 <ul>
 <li>the 99-dimensional integral homology of SL2(Z[1/7]) is
H99(SL2(Z[1/7]),Z) = Z4 + Z12.
(This homology was first calculated by <a
href="http://arxiv.org/abs/math/9503230">A. Adem and N. Naffah</a>). 
 </li>
 <li>the 4-dimensional integral homology of SL3(Z) is
H4(SL3(Z),Z) = Z2. </li>
 <li>the 6-dimensional integral homology of the Bianchi group SL2(Z[w])
with w2=-2 is H6(SL2(Z[w]),Z) = Z2.
 
 </li>
 </ul>
 <ul>
 <li>the classical braid group B on eight strings
(represented by a linear Coxeter diagram D with seven vertices) has
5-dimensional integral homology H5(B,Z) = Z3 .</li>
 <li>the amalgamated product G=S5*AS4
of the symmetric groups S5 and S4 over the
canonical subgroup A=S3 has 5-dimensional integral homology
H5(G,Z) = (Z2)5 . (The amalgamated
product can be represented as a graph of groups.) 
 </li>
 <li>the Heisenberg group H in five complex variables (a torsion
free nilpotent group of class two) has 5-dimensional integral homology H5(H,Z)
= (Z2)43+Z6+Z132.</li>
 <li>the free nilpotent group N of class 2 on four generators
has 4-dimensional integral homology H4(N,Z) = (Z3)4+Z84.
(The <a href="aboutExtensions.html#Lambe">full range</a> of homology
groups
for N were first calculated in a paper by L. Lambe.)</li>
 <li>The 3-dimensional crystallographic space group S with
Hermann-Mauguin symbol "P62" has 5-dimensional integral homology H5(S,Z)
= Z2+Z2.</li>
 </ul>
 (The
last three examples require the "AClib", "Polycyclic" and "nq"
packages.
HAP can be loaded without these packages if such examples are not
required..) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionSL2Z(7,100); 
Resolution of length 100 in characteristic 0 for SL(2,Z[1/7]) . 
No contracting homotopy available. 
gap> Homology(TensorWithIntegers(R),99); 
[ 4, 12 ] 
 
 
gap>
C:=ContractibleGcomplex("SL(3,Z)");; 
gap> R:=FreeGResolution(C,5);; 
gap> Homology(TensorWithIntegers(R),4); 
[ 2 ] 
 
 
gap> C:=ContractibleGcomplex("SL(2,Z[sqrt(-2)])");; 
gap> R:=FreeGResolution(C,7);; 
gap> Homology(TensorWithIntegers(R),6); 
[ 2 ] 
 
 
gap>
D:=[ [1,[2,3]], [2,[3,3]], [3,[4,3]],
[4,[5,3]], [5,[6,3]], [6,[7,3]] ];; 
gap> CoxeterDiagramDisplay(D);; 
 <div style="text-align: center;"><img alt="" src="cd.gif"
style="width: 150px; height: 115px;"> 
 </div>
gap> GroupHomology(D,5);time; 
[ 3 ] 
13885 
 
 
 
gap> S5:=SymmetricGroup(5);SetName(S5,"S5"); 
gap> S4:=SymmetricGroup(4);SetName(S4,"S4"); 
gap> A:=SymmetricGroup(3);SetName(A,"S3"); 
gap> AS5:=GroupHomomorphismByFunction(A,S5,x->x); 
gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x); 
gap> D:=[S5,S4,[AS5,AS4]]; 
gap> GraphOfGroupsDisplay(D); 
 <div style="text-align: center;"><img alt="" src="graphgroups.gif"
style="width: 172px; height: 90px;"> 
 </div>
gap> GroupHomology(D,5);time; 
[ 2, 2, 2, 2, 2 ] 
22004 
 
 
gap> GroupHomology(HeisenbergPcpGroup(5),5);time; 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 
 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0 ] 
73765 
 
 
gap> F:=FreeGroup(4);; N:=NilpotentQuotient(F,2);; 
gap> GroupHomology(N,4);time; 
[ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 
41967 
 
 
gap> GroupHomology(SpaceGroupBBNWZ("P62"),5);time; 
[ 2, 2 ] 
4336 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
command GroupHomology(G,n)
is a composite of several more basic HAP functions and
attempts, in a fairly crude way, to make reasonable choices for a
number of parameters in
the calculation of group homology. For a particular group G you would
almost
certainly be better off using the more basic functions directly and
making the
choices yourself! Similar comments apply to functions for cohomology
(ring) calculations. 
 
The subsequent pages of this manual explain the basic HAP functions. The
intending reader should be aware that many of the examples are intended
to
illustrate the full potential of HAP and consequently may take
many minutes (and in one or two cases hours) to run. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">For
a given crystallographic space group S the HAPcryst extension (which
uses the Cryst GAP package and the Polymake computational geometry
system) can be used to compute a fundamental cell which tiles euclidean
space in such a way that the tiling is respected by the action of S.
For instance, the following commands compute a fundamental cell for the
3-dimensional space group S with
Hermann-Mauguin symbol "P62" and exhibit the 1-skeleton of this cell. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
fd:=FundamentalDomainStandardSpaceGroup([1/2,1/3,1/5],SpaceGroupBBNWZ("P62"));; 
gap> Polymake(fd,"VISUAL_GRAPH"); 
 
 <div style="text-align: center;"><img
style="width: 300px; height: 300px;" alt="" src="Fundom.png"> 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
end this introduction by mentioning that HAP can also be used to make
calculations such as: 
 <ul>
 <li>The rank of the mod 2 cohomology group Hk(M11,Z2)
of the Mathieu group M11 of order 7920 is equal to the
coefficient of xk in the Poincare series p(x) = (x4-x3+x2-x+1)/(x6-x5+x4-2x3+x2-x+1)
for all k less than 15. (This Poincare series for the ring H*(M11,Z2)
was first calculated in [P.Webb, "A local method in group cohomology", Comm. Math. Helv. 62 (1987)
135-167]. ) </li>
 <li>The mod 2 cohomology ring H*(D32,Z2)
for the dihedral group of order 64 is generated by two elements of
degree 1 and one element of degree 2 and possibly (though not very
likely)
some generators of degree greater than 30. 
 </li>
 <li>The Lie algebra M3(Z) of all integer 3×3
matrices has 5-dimensional Lie homology H5(A,Z)=(Z2)8+Z.</li>
 <li>The suspension X=SK(G,1) of an Eilenberg-Mac
Lane space for the free nilpotent group G of class 2 on four generators
has third homotopy group
pi3X = Z30 .</li>
 <li>The double suspension Y=SSK(G,1) of an Eilenberg-Mac Lane
space for the group G=GL(4,3) of 4×4 matrices over the field of
three elements (of order 24261120) has fourth homotopy group pi4Y
= Z2 . 
 </li>
 <li>The free nilpotent Lie algebra A of class two on four
generators, over
the ring of integers Z, has 3-dimensional Leibniz homology HL3(A,Z)=(Z2)8
+ (Z6)16 +Z176 .</li>
 <li>The group presentation P = <x,y,z,a,b,c, | a=xy,
b=yz, c=zx, ax=ya, by=zb, cz=xc > is aspherical.</li>
 <li>The 3-dimensional module M over the field F of two
 elements, arising from the canonical left action of the group G=Syl2(GL3(2))
of 3×3 matrices (of order 8), has a 6-dimensional fifth Ext
module Ext5FG(M,F)=F6.</li>
 <li>The 3-dimensional integral homology of the
homotopy 2-type X represented by the automorphism crossed module D16
--> Aut(D16) is H3(X,Z)=Z2+Z2+Z4.
 
 </li>
 </ul>
The following commands yield these seven calculations. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
PoincareSeriesPrimePart(MathieuGroup(11),2,14); 
(x^4-x^3+x^2-x+1)/(x^6-x^5+x^4-2*x^3+x^2-x+1) 
 
gap> H:=ModPCohomologyGenerators(DihedralGroup(64),30);; 
gap> List(H[1], H[2]); 
[ 0, 1, 1, 2 ] 
 
gap>
A:=MatLieAlgebra(Integers,3);; 
gap> LieAlgebraHomology(A,5); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 0 ] 
 
gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);; 
gap> ThirdHomotopyGroupOfSuspensionB(G); 
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
 0, 0, 0, 0, 0 ] 
 
gap> G:=Image(IsomorphismPermGroup(GL(4,3)));; 
gap> NonabelianSymmetricKernel_alt(G); 
[ [ ], [ 2 ] ] 
 
gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);; 
gap> L:=LowerCentralSeriesLieAlgebra(G);; 
gap> LeibnizAlgebraHomology(L,3); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
6, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0 ] 
 
gap>
F:=FreeGroup(6);;x:=F.1;;y:=F.2;;z:=F.3;;a:=F.4;;b:=F.5;;c:=F.6;; 
gap> rels:=[a^-1*x*y, b^-1*y*z, c^-1*z*x, a*x*(y*a)^-1,
b*y*(z*b)^-1, c*z*(x*c)^-1];; 
gap> IsAspherical(F,rels);; 
Presentation is aspherical. 
 
gap> M:=GModuleByMats(GeneratorsOfGroup(SylowSubgroup(GL(3,2),2)),GF(2));; 
gap> R:=ResolutionFpGModule(DesuspensionMtxModule(M),5);; 
gap> Cohomology(HomToIntegersModP(R,2),4); 
6 
 
gap>
C:=AutomorphismGroupAsCatOneGroup(DihedralGroup(32));; 
gap> N:=NerveOfCatOneGroup(C,4);; 
gap>
K:=ChainComplexOfSimplicialGroup(N);; 
gap> Homology(K,3); 
[ 2, 2, 4 ] 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;"> 
 <table style="width: 100%; text-align: left;" border="0"
cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="text-align: left; vertical-align: top;">
 
 </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="aboutDefinitions.html">Next page</a></td>
 </tr>
 </tbody>
 </table>
 
 </td>
 </tr>
 </tbody>
</table>
 
 
</body>
</html>

Messung
C	H	Z	G
88	99	100	95

¤ Dauer der Verarbeitung: 0.4 Sekunden (vorverarbeitet am 2026-11-04) ¤

Wurzel

Suchen

PVS Prover

Isabelle Prover

Cephes Mathematical Library

Vienna Development Method

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.