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 <td style="vertical-align: top;"><a
href="aboutDavisComplex.html">Previous</a> 
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Functorial Properties 
 </big></td>
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href="aboutCoefficientSequence.html">next</a> 
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 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Homology
is
a
functor.
That is, for any n>0 and group homomorphism f : G → G' there is an
style="font-style: italic;">induced
homomorphism Hn(f) : Hn(G,Z) → Hn(G',Z)
satisfying 
 <ul>
 <li>Hn(gf) = Hn(g)Hn(f) for
group homomorphisms f : G → G', g : G' → G".
 <li>Hn(f) is the identity homomorphism if f is the
identity.</li>
 </ul>
 The following
commands compute H3(f) : H3(P,Z) → H3(S5,Z)
for
the
inclusion
f : P → S5 into the symmetric group S5
of its Sylow 2-subgroup. They also show that the image of the induced
homomorphism H3(f) is
precisely
the
Sylow
2-subgroup of H3(S5,Z). 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
S_5:=SymmetricGroup(5);;
P:=SylowSubgroup(S_5,2);; 
 
gap> f:=GroupHomomorphismByFunction(P,S_5, x->x);; 
 
gap> R:=ResolutionFiniteGroup(P,4);; 
 
gap> S:=ResolutionFiniteGroup(S_5,4);; 
 
gap> ZP_map:=EquivariantChainMap(R,S,f);; 
 
gap> map:=TensorWithIntegers(ZP_map);; 
 
gap> Hf:=Homology(map,3);; 
 
gap> AbelianInvariants(Image(Hf)); 
[2,4] 
 
gap> GroupHomology(G,3); 
[2,12] 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
above computation illustrates a general result. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="10" cellspacing="2">
 <tbody>
 <tr>
 <td
style="text-align: left; background-color: rgb(204, 255, 255); vertical-align: top;">For any Sylow p-subgroup P of a finite
group G, the p-primary part of Hn(G,Z) is a quotient of Hn(P,Z). </td>
 </tr>
 </tbody>
 </table>
 
We denote by Hn(G,Z)(p) the p-part of Hn(G,Z). This result follows from
a property of the transfer
homomorphism 
 
 
 <div style="text-align: center;">Tr(G,K) : Hn(G,Z) → Hn(K,Z)

 
 </div>
 
which exists for any group G and subgroup K<G of finite index |G:K|,
and any n>0. The relevant property is the following. 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left; color: rgb(0, 0, 102);"
border="0" cellpadding="10" cellspacing="2">
 <tbody>
 <tr>
 <td
style="background-color: rgb(204, 255, 255); vertical-align: top;">The
composed homomorphism 
 
 <div style="text-align: center;">Hn(K<G)oTr(G,K)
:
Hn(G,Z) → Hn(G,Z) 
 </div>
 
is just multiplication by the index |G:K|. 
 </td>
 </tr>
 </tbody>
 </table>
 
So in the case when K is a Sylow p-subgroup the composed homomorphism
is an
isomorphism on Hn(G,Z)(p) and, consequently, the
induced homomorphism Hn(K→G,Z) must map surjectively
onto Hn(G,Z)(p). 
 
Another consequence of the transfer (with K=1) is that the exponent of Hn(G,Z)
divides
the
order
|G| for any finite group G. In particular, Hn(G,Z)
is finite (since it is readily seen to be a finitely
generated abelian group). 
 
These results suggest that the homology Hn(G,Z) of a large
finite group G (such as the Mathieu group G=M23) should be
calculated by computing its p-part Hn(G,Z)(p)
for each prime p dividing |G|. For a Sylow p-subgroup P there is a nice
description of the kernel of the surjection Hn(P,Z) → Hn(G,Z)(p).
It
is
generated
by
elements 
 
 <table
style="margin-left: auto; margin-right: auto; width: 80%; text-align: left;"
border="0" cellpadding="12" cellspacing="2">
 <tbody>
 <tr>
 <td
style="background-color: rgb(204, 255, 255); text-align: center; vertical-align: top;">hK(a)
- hxKx-1(a) 
 </td>
 </tr>
 </tbody>
 </table>
 
where x ranges over the double coset representatives of P in G, K is
the intersection of P and its conjugate xPx-1, the
homomorphisms hK, hx-1Kx:Hn(K,Z)
→
Hn(P,Z) are induced by the inclusion K→P,
k→k and the conjugated inclusion K→P, k→x-1kx,
and a ranges over the generators of Hn(K,Z). 
 
The function PrimePartDerivedFunctor(G,R,T,n)
uses this
description of the kernel to compute the abelian invariants
of Hn(G,Z)(p) starting from: 
 <ul>
 <li>the group G, </li>
 <li>a ZP-resolution R for a Sylow p-subgroup P<G, </li>
 <li>the functor T=TensorWithIntegers,</li>
 <li>the integer n>0. 
 </li>
 </ul>
The following commands show that the Mathieu group M23 has
third integral homology with 2-part H3(M23,Z)(2)=0.

 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
M_23:=MathieuGroup(23);; 
 
gap> gens:=GeneratorsOfGroup(SylowSubgroup(M_23,2));; 
 
gap> gensP:=[gens[4],gens[6],gens[2]];; #This list
generates a Sylow 2-subgroup of M_23. 
 
gap> R:=ResolutionFiniteGroup(gensP,4);; 
 
gap> T:=TensorWithIntegers;; 
 
gap> PrimePartDerivedFunctor(M_23,R,T,3); 
[ ] 
 </td>
 </tr>
 <tr>
 <td
style="text-align: left; background-color: rgb(255, 255, 255); vertical-align: top;"><a
name="Milgram"></a>Similar
commands can be used to show that Hn(M23,Z)=0 for
n=1,2,3,4. The triviality of the first four integral homology
groups of M23 was first proved in [J. Milgram, J. Group
Theory, 2000] and answered a conjecture of J.-L. Loday. A group G is
said to be k-connected if Hn(G,Z)=0
for
n=1,
2,
..., k. Back in the mid 1970s Loday had asked if the
trivial group is the only 3-connected finite group. 
 
No example of a 5-connected finite group is yet known! A group is said
to be superperfect if it is
2-connected. A list of some superperfect groups, together with their
third integral homology, is given <a style="font-weight: bold;"
href="aboutSuperperfect.html">here</a>. 
 
The higher dimensional integral homology of a group G is readily
calculated by this method when G has no large Sylow subgroup. For
example, the following commands show that the symmetric group of degree
5 has 20-dimensional integral homology H20(S5,Z)
= (Z2)7 . (We could of course have incorporated
into our computation the fact
that cyclic Sylow groups have trivial integral homology in even
dimensions.) 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
S_5:=SymmetricGroup(5);;

 
 
gap> P2:=SylowSubgroup(S_5,2);; 
 
gap> P3:=SylowSubgroup(S_5,3);; 
 
gap> P5:=SylowSubgroup(S_5,5);; 
 
gap> R2:=ResolutionFiniteGroup(P2,21);; 
 
gap> R3:=ResolutionFiniteGroup(P3,21);; 
 
gap> R5:=ResolutionFiniteGroup(P5,21);; 
 
gap> T:=TensorWithIntegers;; 
 
gap> PrimePartDerivedFunctor(S_5,R2,T,20); 
[ 2, 2, 2, 2, 2, 2, 2 ] 
 
gap> PrimePartDerivedFunctor(S_5,R3,T,20); 
[ ] 
 
gap> PrimePartDerivedFunctor(S_5,R5,T,20); 
[ ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Induced
homology
homomorphisms
can
be composed in HAP. For example, the
following commands illustrate the functorial property Hn(gf)
= Hn(g)Hn(f) for the inclusions f:A4→S4,
g:S4→S5 and n=2. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
A4:=AlternatingGroup(4);; 
gap> S4:=SymmetricGroup(4);; 
gap> S5:=SymmetricGroup(5);; 
gap> f:=GroupHomomorphismByFunction(A4,S4,x->x);; 
gap> g:=GroupHomomorphismByFunction(S4,S5,x->x);; 
gap> gf:=Compose(g,f);; 
gap> RA4:=ResolutionFiniteGroup(A4,3);; 
gap> RS4:=ResolutionFiniteGroup(S4,3);; 
gap> RS5:=ResolutionFiniteGroup(S5,3);; 
gap> ef:=EquivariantChainMap(RA4,RS4,f);; 
gap> eg:=EquivariantChainMap(RS4,RS5,g);; 
gap> egf:=EquivariantChainMap(RA4,RS5,gf);; 
gap> tf:=TensorWithIntegers(ef);; 
gap> tg:=TensorWithIntegers(eg);; 
gap> tgf:=TensorWithIntegers(egf);; 
gap> hf:=Homology(tf,2);; 
gap> hg:=Homology(tg,2);; 
gap> hgf:=Homology(tgf,2);; 
gap> elt:=Random(Source(hf));; 
gap> Image(hgf,elt)=Image(Compose(hg,hf),elt); 
true 
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style="color: rgb(0, 0, 102);" href="aboutDavisComplex.html">Previous
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href="aboutContents.html">Contents</a> 
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