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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Poincare series for groups of
order 32 
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
 Poincare series for the
cohomology ring H*(G,Zp) is the infinite series 
 
 <div style="text-align: center;">a0 + a1x +
a2x2 + a3x3 + ... 
 </div>
 
where ak is by definition the dimension of the vector
space Hk(G,Zp) . The Poincare series can be expressed as
a rational
function P(x)/Q(x) where P(x) and Q(x) are polynomials of finite
degree. 
 
The following commands compute Poincare series for all the groups of
order 32. They rely on an algorithm which seems highly unlikely to
produce
a wrong answer. However, there is no
general proof that the algorithm
produces a series which is correct in arbitrarily high degree. Proofs
have to be provided case by case. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
for G in AllSmallGroups(32) do 
> Print("\n Small Group ",IdSmallGroup(G)," has Poincare series \n", 
> PoincareSeries(G),"\n"); 
> od; 
 
Small Group [ 32, 1 ] has Poincare series 
(1)/(-x+1) 
 
Small Group [ 32, 2 ] has Poincare series 
(x^2+x+1)/(-x^5+x^4+2*x^3-2*x^2-x+1) 
 
Small Group [ 32, 3 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 4 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 5 ] has Poincare series 
(1)/(-x^4+2*x^3-2*x+1) 
 
Small Group [ 32, 6 ] has Poincare series 
(1)/(-x^4+2*x^3-2*x+1) 
 
Small Group [ 32, 7 ] has Poincare series 
(x^2-x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 8 ] has Poincare series 
(x^5+x^2+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 9 ] has Poincare series 
(1)/(-x^4+2*x^3-2*x+1) 
 
Small Group [ 32, 10 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 11 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 12 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 13 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 14 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 15 ] has Poincare series 
(1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 16 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 17 ] has Poincare series 
(1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 18 ] has Poincare series 
(1)/(x^2-2*x+1) 
 
Small Group [ 32, 19 ] has Poincare series 
(1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 20 ] has Poincare series 
(x^2+x+1)/(-x^3+x^2-x+1) 
 
Small Group [ 32, 21 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 22 ] has Poincare series 
(1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) 
 
Small Group [ 32, 23 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 24 ] has Poincare series 
(1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 25 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 26 ] has Poincare series 
(x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 27 ] has Poincare series 
(1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) 
 
Small Group [ 32, 28 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 29 ] has Poincare series 
(1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 30 ] has Poincare series 
(x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) 
 
Small Group [ 32, 31 ] has Poincare series 
(1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 32 ] has Poincare series 
(x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) 
 
Small Group [ 32, 33 ] has Poincare series 
(-x^4+x^3+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) 
 
Small Group [ 32, 34 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 35 ] has Poincare series 
(x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 36 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 37 ] has Poincare series 
(1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 38 ] has Poincare series 
(x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 39 ] has Poincare series 
(1)/(-x^3+3*x^2-3*x+1) 
 
Small Group [ 32, 40 ] has Poincare series 
(1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 41 ] has Poincare series 
(x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 42 ] has Poincare series 
(x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 43 ] has Poincare series 
(1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 44 ] has Poincare series 
(x^6+x^5+x^2+x+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 45 ] has Poincare series 
(1)/(x^4-4*x^3+6*x^2-4*x+1) 
 
Small Group [ 32, 46 ] has Poincare series 
(1)/(x^4-4*x^3+6*x^2-4*x+1) 
 
Small Group [ 32, 47 ] has Poincare series 
(x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 48 ] has Poincare series 
(x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 49 ] has Poincare series 
(x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) 
 
Small Group [ 32, 50 ] has Poincare series 
(x^6+2*x^5+3*x^4+3*x^3+3*x^2+2*x+1)/(x^8-2*x^7+2*x^6-2*x^5+ 
2*x^4-2*x^3+2*x^2-2*x+1) 
 
Small Group [ 32, 51 ] has Poincare series 
(1)/(-x^5+5*x^4-10*x^3+10*x^2-5*x+1) 
gap> TimeToString(time); 
"80.006 sec." 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Click
 <a href="aboutPoincareSeriesII.html">here</a> for Poincare series
for the groups of order 64. 
 </td>
 </tr>
 <tr align="center">
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Proving correctness of a poincare series 
 
 <div style="text-align: left;"> 
We list a few simply stated results which, in some cases, suffice to
prove that a computed Poincare series is correct. In these, p(G)
denotes the poincare series of a finite p-group G. We write
p(H)<=p(G) if the i-th coefficient of p(H) is less than or
equal to the i-the coefficient of p(G) for all i. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="2">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);"><big>Theorem 1</big> 
If N is a normal subgroup of G then p(G)<=p(N)p(G/N). 
In particular, if G=N×Q is a direct product then p(G)=p(N)p(Q).</td>
 </tr>
 </tbody>
 </table>
 
This follows immediately from the <a href="aboutExtensions.html">twised
tensor product</a> construction of CTC Wall. 
 
 
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style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="2">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);"><big>Theorem 2 
 </big>If G is a cyclic p-group then p(G)=1/(1-x) 
 </td>
 </tr>
 </tbody>
 </table>
 
So, for example, these two theorems in conjunction with the following
GAP commands prove that the dihedral group G=SmallGroup(32,18) has
poincare series p(G)<=1/(1-x)2 . 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SmallGroup(32,18);; 
gap> N:=NormalSubgroups(G)[5];; 
gap> IsCyclic(N); 
true 
gap> IsCyclic(G/N); 
true 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">To
prove the equality p(G)=1/(1-x)2 we consider the two maximal
elementary abelian subgroups A, B in G and the induced homomorpisms H*(G,Zp)
→ H*(A,Zp) , H*(G,Zp)
→ H*(B,Zp). (Actually, because these
homomorphisms are not yet implemented in HAP we look at the dual
homology homomorphisms: in this case the nth homology is isomorphic to the nth cohomology.) </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
S:=LatticeSubgroups(G);; 
gap> S:=ConjugacyClassesSubgroups(S);; 
gap> S:=List(S,x->ClassElementLattice(x,1));; 
gap> S:=Filtered(S,x->IsElementaryAbelian(x));; 
gap> List(S,x->Order(x)); 
[ 1, 2, 2, 2, 4, 4 ] 
gap> A:=S[5];; 
gap> B:=S[6];; 
 
gap> RG:=ResolutionFiniteGroup(G,4);; 
gap> RA:=ResolutionFiniteGroup(A,4);; 
gap> A2G:=GroupHomomorphismByFunction(A,G,x->x);; 
gap> RA2RG:=EquivariantChainMap(RA,RG,A2G);; 
gap> CA2CG:=TensorWithIntegersModP(RA2RG,2);; 
gap> f1:=Homology(CA2CG,1); 
[ f1, f2 ] -> [ f1, <identity ...> ] 
gap> f2:=Homology(CA2CG,2); 
[ f1, f2, f3 ] -> [ f1, f1*f2*f3, f1*f2*f3 ] 
 
gap> RB:=ResolutionFiniteGroup(B,4);; 
gap> B2G:=GroupHomomorphismByFunction(B,G,x->x);; 
gap> RB2RG:=EquivariantChainMap(RB,RG,B2G);; 
gap> CB2CG:=TensorWithIntegersModP(RB2RG,2);; 
gap> g1:=Homology(CB2CG,1); 
[ f1, f2 ] -> [ f2, <identity ...> ] 
gap> g2:=Homology(CB2CG,2); 
[ f1, f2, f3 ] -> [ f2, f1*f2*f3, f1*f2*f3 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Using
 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="20" cellspacing="2">
 <tbody>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(204, 255, 255);"><big>Theorem 3</big> 
Let E be the elementary abelian group of order 2n. Then the
graded cohomology ring H*(E,Z2) is isomorphic to
the
free polynomial ring Z2[x1,x2, ... ,xn]
in which each xi has degree 1. 
 </td>
 </tr>
 </tbody>
 </table>
 
we have H*(A,Zp) = Zp[x,y] and H*(B,Zp)
= Zp[u,v] . From the above HAP computations we can
deduce that the image of the induced ring homomorphism 
 
 <div style="text-align: center;">f+g: H*(G,Zp)
→ H*(A,Zp) + H*(B,Zp)
= Zp[x,y] + Zp[u,v] =Zp[x,y,u,v]/<xu=0,xv=0,yu=0,yv=0] 
 <div style="text-align: left;"> 
contains the polynomials x, u, xy+uv and their products such as: 
 
 
 <table style="width: 100%; text-align: left;" border="1"
cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;">Degree 
 </td>
 <td style="vertical-align: top;">Number 
of 
Polynomials 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">0 
 </td>
 <td style="vertical-align: top;">1 
 </td>
 <td style="vertical-align: top;">1 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">1 
 </td>
 <td style="vertical-align: top;">2 
 </td>
 <td style="vertical-align: top;">x 
 </td>
 <td style="vertical-align: top;">u 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">2 
 </td>
 <td style="vertical-align: top;">3 
 </td>
 <td style="vertical-align: top;">x2</td>
 <td style="vertical-align: top;">u2</td>
 <td style="vertical-align: top;">xy+uv</td>
 <td style="vertical-align: top;"> 
 </td>
 <td style="vertical-align: top;"> 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">3 
 </td>
 <td style="vertical-align: top;">4 
 </td>
 <td style="vertical-align: top;">x3</td>
 <td style="vertical-align: top;">u3</td>
 <td style="vertical-align: top;">x2y</td>
 <td style="vertical-align: top;">u2v</td>
 <td style="vertical-align: top;"> 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">4 
 </td>
 <td style="vertical-align: top;">5 
 </td>
 <td style="vertical-align: top;">x4</td>
 <td style="vertical-align: top;">u4</td>
 <td style="vertical-align: top;">x3y</td>
 <td style="vertical-align: top;">u3v</td>
 <td style="vertical-align: top;">x2y2+u2v2</td>
 </tr>
 </tbody>
 </table>
 
 Hence the poincare series for the dihedral group satisfies
p(G) >= 1+2x+3x2+4x3+5x4+... =
1/(1-x)2. This completes the proof that p(G) = 1/(1-x)2
 . 
 </div>
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