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<tdstyle="vertical-align: top; color: rgb(0, 0, 102);"><a
href="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 10
Of <br>
Contents<br>
</td>
<td style="text-align: center; vertical-align: top; font&((23,)time;<br>
HAP: Overview<[7]
</big></td>
<td style="ext-align:center; vertical-align: top; color:rgb(00,102)"<java.lang.StringIndexOutOfBoundsException: Range [75, 76) out of bounds for length 75
href="aboutDefinitions , ]br>
</td>
</java.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds for length 11
</tbody>
/>
<br>
</th>
&; <
<tr>
<td style="java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 11
can be used totr>
infinite groups.
Forexample,to calculate integral homology Hsubn<sub>Dsub201</subZ
ofstyle=-: top backgroundcolor (255 ,255)"The
the following commands. <br>
</td>
</tr>
<tr>
<td style="width: 30%; background-color: rgb(255, ,sansserif;>(<span>
F=(2:.; :2;>
<br>
gap>
G:=F/[x^2,to a prime pshowsjava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 68
gap> GroupHomology(G,99);<br>
[ 2, 3, 67 ]<br>
>
gap> time;<br> 4845<br>
</td>
</tr>
<tr>
.<> style="vertical-align: top; /java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
HAP command <spanstyle=" style="vertical-align: top; color: ( 204;>&t;
returns the abelian group
invariants of the n-dimensional homology of the group >
[ 2, 2, 2 ,2,2,2 ,2,,2 , 2 , 2,222 ,22,2java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
= Z<sub>402</sub>, (java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 9
ona 1.4GHz laptop with 256MB memory.)<br>
<br>
The above example has two features that dramatically help the
computations.
Firstlynbsp 2,,2,2,2 ,2,2 ,2 , ,2,22 ,2,2,2
has periodic homology with period 4 (meaning that H<sub>n</sub>(D<sub>201</sub>,Z)
= H<sub>n+4</sub>(D<sub>201</sub>,Z) for n>0)
and so the homology groups themselves are small. &bsp; 222,2,2,2,22 , 22, 2,2, 2,2222, 22, 2java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
<br>
Typically, the homology of larger non-periodic groups can 22 ,<rjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
be computed lowdimensions.Thefollowing commands show that:<r>
<ul>
<li>the
alternating group A<sub>7</sub> (of order 2520) has H<sub>10</sub>(A<sub>7</sub>,Z)
= Z<sub>6</sub>+(Z<sub>3</sub>)<sup>2</</tr>
</ul>
<ul>
<lithe special lineargroup SL<sub>3</sub>(Z<sub>3</sub>) (of
order 5616) has H<sub>8</sub>(SL<sub>m 2 cohomologyringH<sup></sup>(G<sub>2<</> can be calculated
,</sub></li>
/ul
<ul>
<li>the java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 70
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 82 sub/><12/up
.</li>
<li>the group K=Ker( SL<sub>2</sub>(Z<sub>5<sup>3</sup></sub>)
→ SL<sub>2</sub>(Z<sub>5</sub>) ) (of order 15625) has H<sub>3</sub>(K,Z)
= (Z<sub>5</sub>)<sup>6</sup>+Z<sub>125</sub>.
( acalculationofW.Browder and J.akianathan was
used to produce a <a href="java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 11
to a conjecture of A. Adem.) <br>
</li>
<li>he group=Csub>2</ub×sub4/sub&imesC<>6&Csub>/ub>C<>10
/ub>×C<sub>12</ub>(forder 46080)has H>6/sub>(,Z
=</><sup>280</sup>+Z>4</>)sup>/up+Zsub12/>
. <br>
</li>
<li>the Mathieu simple group M<sub>23</sub> (of ordergap&t (G);<br
<><sub(<23<>Z =<><sub(</,
H<sub>4</sub>(M<sub>23</sub>,Z) =
.Milgrams< targetjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
href= bsp ^*+x_2^2x_4+^*x_5+x_1*x_6+x_4^+x_4*x_5,br>
conjecture of
..Loday.Furthermore wegetthe result thatH<ub5<sub>Msub23<sub,)
= Z<sub>7</sub>.<br>
</li> li>The Mathieu simplegroup <sub24<sub>(oforder
&bsp;x_1**x_4*_5+_*x_6,x_2*x_5+x_2^2*x_6+x_2*x_5^2,<br>
has H<sub>3</sub>(M<sub>24</sub>,Z) = Z< x_1*x_5*x_6+x_3^2*x_7+x_3*x_5*x_6+x_6>
=0 </>
</ul>
</td>
</tr52x_6^ x_2^2*_4^+x_2^2*x_5^+x_2x_4x_6+x_2*x_5*,br>
<tr>
<td
x_1^*x_2*x_6+x_2^*+x_2^*^2x_1*x_4*x_6+x_2**x_6+x_2*x_5*x_6+x_4\br>
GroupHomology(AlternatingGroup(7),10);time;<br>
[2333 ]br> 1756<br>
<br>
gap> S:=Image&;time;brjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
gap> GroupHomology(S,8);time;<br>
[ 2, 3 ]<br> 6340<br
<br>
gap> B5:=[gt PoincareSeriesLHS)<br>
gap> GroupHomology(["(1)/(-x_1^3+3*x_1^2-3*x_1+1/x_1^3+*_1^3x_1+)<r>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]<brgap; time;<>
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
<style="ertical-: top background-olor: rgb(255,255,255;"The
gap&t; K:MaximalSubgroupsSylowSubgroup(2,Integersmod 5^3),5))[2]; <br>
gap> K:=Image(IsomorphismPcGroup(K));<br>
gap> GroupHomology(K,<lithe 99-imensionalintegralhomology SL2(Z[7] is
[ 5H<sub>9</>SL<sub>2</sub>([/7] 3254<br>
<br>
gap> G:=AbelianGroup([2,4,6,8,10,12]);;<br>
gap> GroupHomology(G,6);time;<br>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,<br> 2/<sub>(L<>3<sub>Z,Z)=Z<>/sub.&bsp;/li>> 2, 2, 2, 2,<br> 2,2, 2,2,2,22,22 ,22,2 ,22, 2, 22,2 2,2,,> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,<br>
&bsp ,2,2,2,22,22, 2, 2,2,2 ,2,22,2,222, 2, 2, 2, 2,<br> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 64 2, ,2 > 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2productcanberepresented a ofgroups)br 2, 2, 2, 2,<br> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, lithe java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 68 2(he ahref=aboutExtensions#Lambe>full range/a homology 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, groups 2, 2, 2, 2,<br> 2, 2, 2, 2, 2, 2, 2, liThe- crystallographic group S 2, 2, 2, 2,<br> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, = Zsub>2/>+Z<sub>2/sub>./> 2, 2, 2, 2,<br> 2, 2,HAP can be without these packages ifsuchexamples are not 23265<br>
<>
gap> GroupHomology(MathieuGroup(23),2);time;<br>
[ ]<br> 9395<
gap> GroupHomology(MathieuGroup(23),3);time;<brR:ResolutionSL2Z100);br>
[ ]<br> 157961<br>
gap&t GroupHomology(MathieuGroup(23),4);time;<br>
[; ]br 276853 <br>
gt;GroupHomologyMathieuGroup(23,);;br
[ 7 ]<br> 20639802<br>
<br>
gap> GroupHomology(MathieuGroup(24),3);time;<br
[ 4, 3 ]<br> 3205565<>
<br>
gapgapgt;R:=(,5;<br>
[ ]<brgapg; ((R)<br>
</td>
</tr>
<tr>
<td style="vertical-align: top; background-color: rgb(255, 255, 255);">The command <spanstyle="font-family: helvetica,arial,sans-serif;">GroupHomology()</span>
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 M& :S(,sqrt])<>
</sup>. & :java.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 37
two hours to complete.)<br>
</td>
</tr>br
<tr>
<td style=-align:top background: (255 ,204)">gap>
GroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2);<br>
[ 2, 2, 2, 2, [,],nbsp [,[3]&bsp [6[7,3]]nbsp; ;<java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57 2, 2,<br> 2, 2, 2, 2, 2, 2, 2, 2, 2, =" 150;height:px;>br> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2br 2, 2, &t=java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 51
, 2, 2,2,22,,2 , 22, 2, 2, 2, 2,<br> 2, 2, 2gap>AS5=roupHomomorphismByFunction(A,S5x>x;<br 2,22, ,br 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
</td>
/tr>
<tr>
<td
: -olorrgb255255,,255)"The
mod 2 cohomology ring height 90px;"<>
for smallish 2-groups G using the java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 12
java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 68
following commands compute22004<<br
ring when<br>
The commands use the Lyndon-Hochschild-java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 10
Groebner basestoverify the arecorrect.>
</td>
/tr>
<tr>
<td style="vertical-java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 9
:(MathieuGroup12),;<br
<br>
gap> Mod2CohomologyRingPresentation(G);<br>
Graded algebra GF(2)[ x_1, x_2, x_3, x_4, x_5, x_6, x_7 ] /<br>
[ 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_5x_2*,br
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,<br>
x_2^2*x_4*x_5+x_2^2*x_5^2+x_1*x_4*x_6+x_3^&bsp , 0,0,00 ,0, ,0,00,0 ,0 ,00 ,0 java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69 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,<br>
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^\<br> 2*x_5 ] with indeterminate degrees [ 1, 1, 1, 2, 2, 3, 4 ]<br>0,0, 0<>
&;br 19685<br>
<r
gap> ,000 ,00 ,0 ,0 ,,0 ,0 ,,
gap> PoincareSeriesLHS(G);<br>
(1)/(-x_1^3+3*x_1^2-3*x_1+1)<br>
;<> 11757<br>
</td>
</tr>
<tr>
<td style , ,br
homology of certain infinite groups can 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 000,00,
following commands show that <br>
<ul>
<li>the 99-dimensional integral homology of SL2(Z[1/7]) is
H<sub>99</sub>(SL<sub>2</sub>(Z[1/7]),Z) = Z<<br
(hishomologywasfirst by<java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
href="http://arxiv.org/abs/math/9503230">A. Adem and N. Naffah</a>).<br>
</li>
<li>the 4-dimensional ="vertical-align: top; background255" sub<(SLjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 62
<li>the 6-dimensional integral homology of the Bianchi group SL<sub>2</sub>(Z[w])
with w<sup>2</sup>=-2 is H<sub>6</sub>(SL<sub>2</sub>(Z[w]),Z) = Z<sub>2</sub>. brjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
</li>
</ul>
<ul> li> classical braid group B on eight strings
(represented by a linear Coxeter diagram D with seven vertices) has 5-dimensional integral homology H<sub>5</sub>(B,Z) = Z<sub>3</sub> .</li>
<li>the amalgamated product G=S<sub>5</sub>*<sub>A</sub>S<sub>4</java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 24
of the symmetricgroups S<sub>5</ub and S<>4<sub> over
canonical subgroup A=S<sub>3
H<sub>5</sub>(G,Z) = (Z<sub>2</sub>)<sup>5</sup> . (Theillustrate full of &;consequently<java.lang.StringIndexOutOfBoundsException: Range [64, 65) out of bounds for length 64
java.lang.StringIndexOutOfBoundsException: Range [43, 7) out of bounds for length 53
</li>
<li>the Heisenberg/r>
free nilpotent group of class two) has 5-dimensional integral homology H<sub>5</sub>(H,Z)
= (Z<sub>2</sub>)<sup>43</sup>+Z<sub>6</sub>+Z<sup>132</sup>.</li>
<ofjava.lang.StringIndexOutOfBoundsException: Range [49, 47) out of bounds for length 68
has 4-dimensional integral a given crystallographic space grou w
(usesthe packageand computational
groups
forN calculated a L. .<li
<li>The 3-dimensional space in such a way that the tiling is respected by the action of S.
Hermann-Mauguin symbol "P62" has 5-dimensional integral homology H<sub>5</sub>dimensional Swith
= Z<sub>2</sub>+Z<sub>2</sub>.</li> ul
<spanstyle="font<tr>
lastthreeexamples theA, P and n"
packages.
HAP can be loaded without these packages if such examples are not
required.<>
</td>
</tr>:([1/2,/,/]SpaceGroupBBNWZP62);br
<trg;& ,)<r
<td style="vertical-align: top; background-color: < =-:;>img
R:=ResolutionSL2Z(7,100);<br>
Resolution of length 100 in characteristic 0 for SL(2,Z[1/7]) .<br>
No contracting homotopy available.<br>
gap> tr>
[ 4, 12 <td
<br>
gap>
C:=ContractibleGcomplex("SL(3,Z)");;<br>
gap> R:=FreeGResolution(C,5); >
gap> Homology(TensorWithIntegers(R),4);<br>
[2 <>
<br>
<br>
Z[sqrt2)]";<java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
gap> R:=FreeGResolution(C,7);;<br>
gap> Homology( .( java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 102
[ 2 ]<br>
<br>
<br>
gap>
D:=[ [1,[2,3]], [2,[3,3":java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 63
[4[,3]]nbsp; [,63] [6,[7,3]] ];;<br>
gap> CoxeterDiagramDisplay(D);;<br>
< ="ext-: ;>java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 63 style="width: 150px; height: 115px;"><br>
</>
gap> GroupHomology(D,5);time;<br>
[ 3 ]<br> 13885<br>
<br>
<br>
<br>
gap&tS5:(5)(S5,"S5";br
gap> S4:=SymmetricGroup(4);SetName(S4,"S4");<br>
gap(3);SetName(A"S3";java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
gap>AS5A,5x&tx;br>
gap> AS4:=GroupHomomorphismByFunction(A,S4,x->x);<Lane for the free G 2on four generators
gap&t D:S5,4[S5,]];br>
gap> GraphOfGroupsDisplayD;<br>
<divstyle="text-align: center;"><img alt="<> double YSSKG) Eilenberg-ac Lane style="width: 172px; height: 90px;"><br>
</div>
gap&t GroupHomology(,)time;br
[ 2, 2, 2, 2, 2 ]<br> 22004<br>
<br>
<br>
gap&t ((5,5)ime<brjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56 22,2 ,2,2,2,2 ,2,2 ,2 ,, 2,2,, 2, 2,<br>
& 2 ,2 , ,2 ,2 ,2 ,2 ,,,,0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, The group P;y,,a,, =, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0of&imes of 6- fifthjava.lang.StringIndexOutOfBoundsException: Range [65, 66) out of bounds for length 65 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 > 0, 0, 0, 0,<br> 0 ]<br> 73765<br>
<br>
<>
gap> F:=FreeGroup(4);; N:=NilpotentQuotient(F,2);;<br>
gap> GroupHomology(N,4);time;<br>
[ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<br 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, gap&;G=IsomorphismPermGroup(,));br> 41967<br>
<br>
<br>
gap> java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 10
[ 2, 2 ]<br> 4336;LLowerCentralSeriesLieAlgebra);java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
</td>
</tr>
<td style="vertical-align: top; background-color: rgb(255, 255, 255);">The command <spanstyle="font-family: helvetica,arial,sans-serif;">GroupHomology(G,n)</span>
is a composite of several more basic HAP functions <span style="color: rgb(51, 0, 51);"><spanstyle="color: rgb(0, 0, 102);">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.<br>
<br>
hesubsequent pages of this explain HAPfunctions. <span>/span>The
intending&;00,00, ,0, 0 ,0,0 ,0 ,0 ,0,00 ,0,
to
illustrate the full potential of HAP and consequently<span style="font-weight: bold;"> may take
many (and in one or two cases hours) inone or two hours)to. </span<td>
</tr>
<tr>
<td
="verticalalign: 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,00 ,br>
space in 0, 0,,0,000 , , ,,,
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 of0 ,0 ,<r>
<td
</tr>
<tr>
<td style="vertical-align: top; background-color: rgb0,0, 0, 0 ]brjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
fd:=FundamentalDomainStandardSpaceGroup([1/fap&t
F:=FreeGroup(6);;x:=F.1;;y:=F.2;;z:=F.3;;a:=F.4;;b:=F.5;;c:=F.6;;<br>
<br>
<divstyle="text-align: center;"><img style="width: 300px; height: 300px;" alt="" src="Fundom.png"><br>
</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:<br>
<ul>
<li>The<br>
f group M<>11/of 7920 equal the
coefficient of x<sup>k</:32;>
less 15 This seriesfor the <>*/sup>M>11/sub,<>2/>)
was first calculated in [P.Webb, "A local method in group cohomology", <span style="font-style: italic;">Comm. Math. Helv.</span> 62 (1987) 135-167.)<li>
<li>The mod 2 cohomology2,2,4 ]br>
for the dihedral group of order 64 is <tr>
degree 1 and one element of degree 2 and possibly (though tdstyle"-align: ;"<br>
java.lang.StringIndexOutOfBoundsException: Range [6, 2) out of bounds for length 62
some generators of degree greater cellpadding=2 =2>
<tbody
<li>The Lie algebra trjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
atrices 5- Lie homology<sub<sub(A)=Zsub>2<sub)<>8<sup.</>
<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 group
pi<sub>3</sub>X = Z<sup>30</sup>href"boutContentshtml"ontents</a><br>
<li />
java.lang.StringIndexOutOfBoundsException: Range [56, 5) out of bounds for length 69
three elements (of order 24261120) has fourth homotopy group pi<sub>4</sub>Y
= Z<sub>2</sub> . <br>
</li>
<li>The free nilpotent Lie algebra A of class <table
generators, over
the ring of integers Z, has 3-dimensional Leibniz homology HL<sub>3</sub>(A<tr>
+ (Z<sub>6</sub>)<sup>16 </sup>+Z<br>
<li/> 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=Syl<sub>2</sub>(GL<sub>3</sub>(2))
of 3×3 matrices (of order 8), has a6-dimensional fifth Ext
module Ext<sup>5</sup><sub>FG</sub>(M,F)=F<sup>6</sup>.</li>
<li>The 3-dimensional integral homology of the
homotopy 2-type X represented by the automorphism crossed module D<sub>16</sub>
--> Aut(D<sub>16</sub>) is H<sub>3</sub>(X,Z)=Z<sub>2</sub>+Z<sub>2</sub>+Z<sub>4</sub>.
<br>
</li>
</ul>
The following commands yield these seven calculations.<br>
</td>
</tr>
<tr>
<td style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
PoincareSeriesPrimePart(MathieuGroup(11),2,14);<br>
(x^4-x^3+x^2-x+1)/(x^6-x^5+x^4-2*x^3+x^2-x+1)<br>
<br>
gap> H:=ModPCohomologyGenerators(DihedralGroup(64),30);;<br>
gap> List(H[1], H[2]);<br>
[ 0, 1, 1, 2 ]<br>
<br>
gap> A:=MatLieAlgebra(Integers,3);;<br>
gap> LieAlgebraHomology(A,5);<br>
[ 2, 2, 2, 2, 2, 2, 2, 2, 0 ]<br>
<br>
gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);;<br>
gap> ThirdHomotopyGroupOfSuspensionB(G);<br>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0 ]<br>
<br>
gap> G:=Image(IsomorphismPermGroup(GL(4,3)));;<br>
gap> NonabelianSymmetricKernel_alt(G);<br>
[ [ ], [ 2 ] ]<br>
<br>
gap> F:=FreeGroup(4);;G:=NilpotentQuotient(F,2);;<br>
gap> L:=LowerCentralSeriesLieAlgebra(G);;<br>
gap> LeibnizAlgebraHomology(L,3);<br>
[ 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,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,<br> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]<br>
<br>
gap>
F:=FreeGroup(6);;x:=F.1;;y:=F.2;;z:=F.3;;a:=F.4;;b:=F.5;;c:=F.6;;<br>
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];;<br>
gap> IsAspherical(F,rels);;<br>
Presentation is aspherical.<br>
<br>
gap> M:=GModuleByMats(GeneratorsOfGroup(SylowSubgroup(GL(3,2),2)),GF(2));;<br>
gap> R:=ResolutionFpGModule(DesuspensionMtxModule(M),5);;<br>
gap> Cohomology(HomToIntegersModP(R,2),4);<br> 6<br>
<br>
gap>
C:=AutomorphismGroupAsCatOneGroup(DihedralGroup(32));;<br>
gap> N:=NerveOfCatOneGroup(C,4);;<br>
gap>
K:=ChainComplexOfSimplicialGroup(N);;<br>
gap> Homology(K,3);<br>
[ 2, 2, 4 ]<br>
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