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 <td style="vertical-align: top;"><a
href="aboutCohomologyRings.html">Previous</a> 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Mod p cohomology rings &
the Steenrod algebra 
 </big></td>
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href="aboutTwistedCoefficients.html">next</a> 
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">There
is
much
interest
in
the
ring
structure
of
the
mod p cohomology H*(G,Zp)
of
p-groups
G. 
 <ul>
 <li>Jon Carlson has computed these rings for all groups of
order 64, the smaller 2-groups having been handled previously. He has
contributed some very efficient functions to the <a
href="http://magma.maths.usyd.edu.au/magma/">Magma</a> computer
algebra system. His methods are described <a href="carlson.pdf">here</a>,
and
his
calculations
can
be
found

 <a href="http://www.math.uga.edu/%7Elvalero/cohointro.html">here</a>.
His
methods
rely
on
matrix
algebra
over the field of p elements and are not
directly suitable for large groups.</li>
 <li>David Green has developed non-abelian Gröbner basis
methods for calculating the mod p homology of larger groups. He has
implemented his methods as a stand-alone C package. His calculations
can be found <a
href="http://www.math.uni-wuppertal.de/%7Egreen/Coho/index.html">here</a>.</li>
 <li>Marcus Bishop has implemented Jon Carlson's functions in
the GAP package <a href="https://www.gap-system.org">Crime</a>.</li>
 </ul>
Since the GAP package Crime focuses on mod p cohomology, only a few
functions on this topic have so far been included in HAP. 
 
At present these HAP functions work differently to those for integral
cohomology in that they rely heavily on matrix algebra and minimal
resolutions. More work needs to be done on improving the effeciency of
these functions. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Let
G be the group G=SmallGroup(64,135) in the small groups
library. The following HAP commands compute the ring H*(G,Zp)
modulo
all
elements
of
degree
greater
than
10.
The
ring is returned as
a structure constant algebra A over the field of two elements. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
R:=ResolutionPrimePowerGroup(SmallGroup(64,135),10); 
Resolution of length 10 in characteristic 2 for <pc group of size 64
with 
6 generators> . 
No contracting homotopy available. 
A partial contracting homotopy is available. 
 
gap> A:=ModPCohomologyRing(R); 
<algebra of dimension 187 over GF(2)> 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
following additional command shows that the ring H*(G,Zp)
is
generated
by
three
elements
in
degree
1,
two
in degree 2,
one in degree 3, one in degree 5, one in degree 8 and possibly some
generators of degree greater than
10. </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
S:=List(ModPRingGenerators(A),A!.degree); 
[ 0, 1, 1, 1, 2, 2, 3, 5, 8 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><a
name="proof"></a>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,Zk) . The Poincare function is a rational
function P(x)/Q(x) equal to the Poincare series. 
 
The following commands compute the Poincare function for the Sylow
2-subgroup of the Mathieu group M12. They rely on an
algorithm which seems unlikely to produce
a wrong answer but for which we have no proof that the answer is
always correct. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">PoincareSeries(MathieuGroup(SylowSubgroup(MathieuGroup(12),2)); 
(1)/(-x^3+3*x^2-3*x+1) </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Poincare
series
for
groups
of
order
32
and
most
of
the groups of order 64 are
listed <a href="aboutPoincareSeries.html">here</a>.</td>
 </tr>
 <tr align="center">
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><big
style="font-weight: bold;"><big>Steenrod Algebra</big></big> 
 
 <div style="text-align: left;">Given a finite p-group G, the
short exact sequence 
 
 <div style="text-align: center;">Z >---> Z --->Zp 
 </div>
 
of ZG-modules induces cohomology connecting homomorphisms b:Hn(G,Zp)
-->
Hn+1(G,Zp) for each n. The collection of all
these connecting homomorphisms is known as the Bockstein homomorphism
b:H*(G,Z_p) --> H*(G,Z_p). The following commands compute the
Bockstein homomorphism on the
generators of the ring H*(G,Zp),
where
p=3 and G=SmallGroup(81,4) is the fourth group of order 81 in GAP's
library of small groups. </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SmallGroup(81,4);; 
gap> A:=ModPSteenrodAlgebra(G,8);; 
gap> List(ModPRingGenerators(A),x->Bockstein(A,x)); 
[ 0*v.1, 0*v.1, v.5, 0*v.1, (Z(3))*v.7+v.8+(Z(3))*v.9 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
Bockstein together with the steenrod squares Sqk:Hn(G,Zp)
-->
Hn+k(G,Zp) for p=2, or steenrod powers Pk:Hn(G,Zp)
-->
Hn+2k(p-1)(G,Zp) for p>2, generate the
Steenrod
algebra. The Steenrod squares and powers are not yet fully implemented. 
 
However, the
following commands compute the Steenrod square Sq1 on the
generators of the ring H*(G,Z2)
where G=SmallGroup(32,8) is the eighth group of order 32 in GAP's
library of small groups. It is known that all mod-2 cohomology
ring generators for this G lie in degrees at most 8, so only the first
nine degrees of the ring
H*(G,Z2)
are
computed. (This is a bit of a cheat because Sq1 is equal to
the Bockstein homomorphism for p=2.) 
 
The higher squares Sqn
are
only partially implemented and return fail
in cases where they are not
yet implemented. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SmallGroup(32,8);; 
gap> N:=9;; 
gap> R:=ResolutionPrimePowerGroup(G,N+1);; 
gap> S:=ResolutionNormalSeries([G,Center(G)],N+1);; 
gap> A:=Mod2SteenrodAlgebra(G,N,R,S);; 
gap> List(ModPRingGenerators(A),x->Sq(A,1,x)); 
[ 0*v.1, 0*v.1, v.4+v.5+v.6, 0*v.1, v.7+v.9+v.10, v.12, v.20,
v.23+v.25, v.28, v.39 ] 
 </td>
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