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 <td style="vertical-align: top;"><a
href="aboutGraphsOfGroups.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Space groups and almost
crystallographic groups - an example of how to piece resolutions
together 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutRosenbergerMonster.html">next</a> 
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 </tr>
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 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
first step in attempting to calculate the cohomology of a group G is to
decide on how best to represent the group. In some cases G will admit a
decomposition into subgroups and quotient groups where the various
component groups need to be represented differently. We try to
illustrate this
with a simple example. 
 
Consider the tessellation of the plane R2
by congruent equilateral triangles, and let G be the group of
isometries of the plane which preserve this tessellation. 
 
 <div style="text-align: center;"><img alt="" src="crystal.jpg"
style="width: 657px; height: 383px;"> 
 </div>
 
The group G is generated by two
translations S,T together with two reflections X,Y. By using the
embedding R2 → R3 (u,v) → (u,v,1)
these affine
transformations can be represented as 3×3 matrices involving the
square root of 3. The following commands construct G. 
 </td>
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 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
x:=Indeterminate(Rationals);; 
gap> p:=x^2-3;; 
gap> K:=AlgebraicExtension(Rationals,p);; 
gap> one:=One(K);; 
gap> rt3:=RootOfDefiningPolynomial(K);; 
 
gap> reflectionX:=[[-1,0,0],[0,1,0],[0,0,1]]*one;; 
gap> reflectionY:=[[1/2,rt3/2,0],[rt3/2,-1/2,0],[0,0,1]]*one;; 
gap> translationS:=[[1,0,2],[0,1,0],[0,0,1]]*one;; 
gap> translationT:=[[1,0,1],[0,1,rt3],[0,0,1]]*one;; 
 
gap> G:=Group([reflectionX,reflectionY,translationS,translationT]);; 
gap> P:=Group([reflectionX,reflectionY]);; 
gap> N:=Group([translationS,translationT]);; 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
linear isometries in G form a finite group P (called the point group). The translations S
and T generate a free abelian group N. The general theory of space
groups tells us that N is normal in G and that the quotient G/N is
isomorphic to P. The groups N and P are created above. The
following command shows that P is the symmetric group of degree 3. 
 </td>
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 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
StructureDescription(P); 
"S3" 
 </td>
 </tr>
 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;">We
can construct a ZP-resolution as
follows. 
 </td>
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
RP:=ResolutionFiniteGroup(P,12);; 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Using
the fact that N is free abelian of rank 2 we construct a ZN-resolution
as follows. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
RN:=ResolutionAbelianGroup([0,0],12);; 
 
gap> fpN:=RN!.group;; 
gap> fpNhomN:=GroupHomomorphismByImages(fpN,N, 

GeneratorsOfGroup(fpN),[translationS,translationT]);; 
 
gap> RN!.group:=N;; 
gap> RN!.elts:=List(RN!.elts,x->Image(fpNhomN,x));; 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
construct a homomorphism G→P, together with a section P→G, as
follows. 
 </td>
 </tr>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
################################ 
gap> GhomPfn:=function(MM) 
> local M,i,j; 
> M:=[]; 
> for i in [1..3] do M[i]:=[]; 
> for j in [1..3] do 
> M[i][j]:=MM[i][j]; 
> od; 
> od; 
> M[1][3]:=0*one; 
> M[2][3]:=0*one; 
> return M; 
> end;; 
gap> ################################ 
 
gap> GhomP:=GroupHomomorphismByFunction(G,P,GhomPfn); 
gap> PmapG:=function(MM); return MM; end; 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
can now combine the ZP-resolution and ZN-resolution into a free
ZG-resolution as follows. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
RG:=ResolutionExtension(GhomP,RN,RP,"Don't Test Finiteness", PmapG);; 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands show that H1(G,Z)=Z2, H2(G,Z)=Z2,
H3(G,Z)=Z3+Z3+Z6, H4(G,Z)=Z2
 and suggest that the homology is periodic with period 4. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
TRG:=TensorWithIntegers(RG);; 
gap> for n in [1..11] do 
> Print("The homology in dimension ",n," is ",Homology(TRG,n),"\n"); 
> od; 
The homology in dimension 1 is [ 2 ] 
The homology in dimension 2 is [ 2 ] 
The homology in dimension 3 is [ 3, 3, 6 ] 
The homology in dimension 4 is [ 2 ] 
The homology in dimension 5 is [ 2 ] 
The homology in dimension 6 is [ 2 ] 
The homology in dimension 7 is [ 3, 3, 6 ] 
The homology in dimension 8 is [ 2 ] 
The homology in dimension 9 is [ 2 ] 
The homology in dimension 10 is [ 2 ] 
The homology in dimension 11 is [ 3, 3, 6] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The following
command shows however that the resolution RG is not periodic with
period 4. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
List([1..11],n->RG!.dimension(n)); 
[ 4, 8, 12, 16, 21, 28, 36, 44, 52, 60, 68 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following command yields the presentation 
 
 <div style="text-align: center;"><s, t, a, y | a3 =
y2 = (ay)2 = sat-1sa-1 =
tasa-1 = (sy)2 = tyt-1sy, tst-1s-1
> 
 
 <div style="text-align: left;">for the infinite group G. 
 </div>
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
PresentationOfResolution(RG); 
rec( freeGroup := <free group on the generators [ f1, f2, f3, f4
]>, 
 relators := [ f4^2, f3^3, f3*f4*f3*f4, f1*f3*f2^-1*f1*f3^-1, 
 f2*f3*f1*f3^-1, f1*f4*f1*f4,
f2*f4*f2^-1*f1*f4, f2*f1*f2^-1*f1^-1 ] ) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
periodic ZG-resolution can be obtained by replacing the above
ZP-resolution by a periodic one. The following additional commands
construct such a periodic ZG-resolution. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FreeGroup(2);; 
gap> relators:=[ F.1^2, F.1*F.2*F.1^-1*F.2^-2 ];; 
gap> RP:=ResolutionSmallFpGroup(F/relators,12);; 
gap> hom:=GroupHomomorphismByImagesNC(RP!.group,P, 

 GeneratorsOfGroup(RP!.group),
[reflectionX,reflectionX*reflectionY]);; 
 
gap> Order(Kernel(hom)); 
1 
gap> Order(Image(hom)); 
6 
gap> # So hom is an isomorphism. 
 
gap> RP!.group:=P;; 
gap> RP!.elts:=List(RP!.elts,x->Image(hom,x));; 
 
gap>
RG:=ResolutionExtension(GhomP,RN,RP,"Don't Test Finiteness", PmapG);; 
 
gap> List([1..11],n->RG!.dimension(n)); 
[ 4, 7, 7, 5, 5, 7, 7, 5, 5, 7, 7 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
above techniques can be applied to any almost crystallographic group G
(that is, a nilpotent-by-finite group G with no non-trivial finite
normal subgroups).
Such groups can be produced using the Cryst and AClib packages. 
 
The function ResolutionAlmostCrystalGroup(G,n)
allows one to construct the resolution directly if G is an almost
crystallographic pcp group. For example, the following commands compute
the ranks of the mod 2 cohomology of a 2-dimensional space group with
point group equal to the cyclic group of order 4. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SpaceGroup(2,10);; 
gap> StructureDescription(PointGroup(G)); 
"C4" 
 
gap> G:=Image(IsomorphismPcpGroup(G));; 
gap> R:=ResolutionAlmostCrystalGroup(G,6);; 
gap> HomR:=HomToIntegersModP(R,2);; 
gap> for n in [1..5] do 
> Print("The mod 2 cohomology in dimension ",n," has rank
",Cohomology(HomR,n),"\n");

> od; 
The mod 2 cohomology in dimension 1 has rank 2 
The mod 2 cohomology in dimension 2 has rank 3 
The mod 2 cohomology in dimension 3 has rank 3 
The mod 2 cohomology in dimension 4 has rank 3 
The mod 2 cohomology in dimension 5 has rank 3 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">An
almost crsystallographic group G has a normal nilpotent subgroup T of
finite index. We define T1=T and Tc+1=[Tc,G].
The command ResolutionAlmostCrystallographicQuotient(G,n,c) produces a
free ZQ-resolution for the group Q=G/Tc. 
 
The following commands calculate the ranks of the mod 2
cohomology of Q=G/Tc for the preceding space group G and
c=2,3,4,5. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
#CASE C=2. 
gap> S:=ResolutionAlmostCrystalQuotient(G,6,2);; Q:=S!.group;; 
gap> Order(Q); 
8 
gap> Coclass(Q); 
2 
 
gap> HomS:=HomToIntegersModP(S,2);; 
gap> for n in [1..5] do 
> Print("The mod 2 cohomology in dimension ",n," has rank
",Cohomology(HomS,n), "\n");

> od; 
The mod 2 cohomology in dimension 1 has rank 2 
The mod 2 cohomology in dimension 2 has rank 3 
The mod 2 cohomology in dimension 3 has rank 4 
The mod 2 cohomology in dimension 4 has rank 5 
The mod 2 cohomology in dimension 5 has rank 6 
 
 
gap> #CASE C=3. 
gap> S:=ResolutionAlmostCrystalQuotient(G,6,3);; Q:=S!.group;; 
gap> Order(Q); 
16 
gap>Coclass(Q); 
2 
 
gap> HomS:=HomToIntegersModP(S,2);; 
gap> for n in [1..5] do 
> Print("The mod 2 cohomology in dimension ",n," has rank
",Cohomology(HomS,n), "\n");

> od; 
The mod 2 cohomology in dimension 1 has rank 2 
The mod 2 cohomology in dimension 2 has rank 4 
The mod 2 cohomology in dimension 3 has rank 6 
The mod 2 cohomology in dimension 4 has rank 9 
The mod 2 cohomology in dimension 5 has rank 12 
 
gap> #CASE c=4. 
gap> S:=ResolutionAlmostCrystalQuotient(G,6,4);; Q:=S!.group;; 
gap> Order(Q); 
32 
gap>Coclass(Q); 
2 
 
gap> HomS:=HomToIntegersModP(S,2);; 
gap> for n in [1..5] do 
> Print("The mod 2 cohomology in dimension ",n," has rank
",Cohomology(HomS,n), "\n");

> od; 
The mod 2 cohomology in dimension 1 has rank 2 
The mod 2 cohomology in dimension 2 has rank 4 
The mod 2 cohomology in dimension 3 has rank 6 
The mod 2 cohomology in dimension 4 has rank 9 
The mod 2 cohomology in dimension 5 has rank 12 
 
gap> #CASE c=5. 
gap> S:=ResolutionAlmostCrystalQuotient(G,6,5);; Q:=S!.group;; 
gap> Order(Q); 
64 
gap> Coclass(Q); 
2 
 
gap> HomS:=HomToIntegersModP(S,2);; 
gap> for n in [1..5] do 
> Print("The mod 2 cohomology in dimension ",n," has rank
",Cohomology(HomS,n), "\n");

> od; 
The mod 2 cohomology in dimension 1 has rank 2 
The mod 2 cohomology in dimension 2 has rank 4 
The mod 2 cohomology in dimension 3 has rank 6 
The mod 2 cohomology in dimension 4 has rank 9 
The mod 2 cohomology in dimension 5 has rank 12 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">A
finite group of order pn and nilpotency class c is said to
have coclass r=n-c. It was
shown by Charles Leedham-Green and others [C.Leedham-Green, The
structure of finite p-groups, J.
London Mathematical Society, (2) 50 (1994) 49-67] that, with a
finite number of exceptions, every p-group of coclass r is associated
to one of only a finite number of p-adic uniserial space
groups G. In particular, for sufficiently large m the groups G/Tm
all have
coclass r. The above calculations are consistent with the conjecture
that almost all groups of coclass r associated to a particular space
group G have "very similar cohomological properties". 
 
For the prime p=2 it is known [J. Carlson,
Coclass and cohomology, J Pure
Applied Algebra 200 (2005) 251-266] that there are only finitely
many isomorphism classes of mod 2 cohomology rings of 2-groups of a
given coclass r. 
 </td>
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style="color: rgb(0, 0, 102);" href="aboutGraphsOfGroups.html">Previous
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 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
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 <td style="text-align: right; vertical-align: top;"><a
href="aboutRosenbergerMonster.html">Next
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