


Homology
is
a
functor.
That is, for any n>0 and group homomorphism f : G → G' there is an induced
homomorphism H_{n}(f) : H_{n}(G,Z) → H_{n}(G',Z)
satisfying


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] 

The
above computation illustrates a general result.
We denote by H_{n}(G,Z)_{(p)} the ppart of H_{n}(G,Z). This result follows from a property of the transfer homomorphism Tr(G,K) : H_{n}(G,Z) → H_{n}(K,Z)
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.
So in the case when K is a Sylow psubgroup the composed homomorphism is an isomorphism on H_{n}(G,Z)_{(p)} and, consequently, the induced homomorphism H_{n}(K→G,Z) must map surjectively onto H_{n}(G,Z)_{(p)}. Another consequence of the transfer (with K=1) is that the exponent of H_{n}(G,Z) divides the order G for any finite group G. In particular, H_{n}(G,Z) is finite (since it is readily seen to be a finitely generated abelian group). These results suggest that the homology H_{n}(G,Z) of a large finite group G (such as the Mathieu group G=M_{23}) should be calculated by computing its ppart H_{n}(G,Z)_{(p)} for each prime p dividing G. For a Sylow psubgroup P there is a nice description of the kernel of the surjection H_{n}(P,Z) → H_{n}(G,Z)_{(p)}. It is generated by elements
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 h_{K}, h_{x1Kx}:H_{n}(K,Z) → H_{n}(P,Z) are induced by the inclusion K→P, k→k and the conjugated inclusion K→P, k→x^{1}kx, and a ranges over the generators of H_{n}(P,Z). The function PrimePartDerivedFunctor(G,R,T,n) uses this description of the kernel to compute the abelian invariants of H_{n}(G,Z)_{(p)} starting from:


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 2subgroup of M_23. gap> R:=ResolutionFiniteGroup(gensP,4);; gap> T:=TensorWithIntegers;; gap> PrimePartDerivedFunctor(M_23,R,T,3); [ ] 

Similar
commands can be used to show that H_{n}(M_{23},Z)=0 for
n=1,2,3,4. The triviality of the first four integral homology
groups of M_{23} 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 kconnected if H_{n}(G,Z)=0
for
n=1,
2,
..., k. Back in the mid 1970s Loday had asked if the
trivial group is the only 3connected finite group. No example of a 5connected finite group is yet known! A group is said to be superperfect if it is 2connected. A list of some superperfect groups, together with their third integral homology, is given here. 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 20dimensional integral homology H_{20}(S_{5},Z) = (Z_{2})^{7} . (We could of course have incorporated into our computation the fact that cyclic Sylow groups have trivial integral homology in even dimensions.) 

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); [ ] 

Induced
homology
homomorphisms
can
be composed in HAP. For example, the
following commands illustrate the functorial property H_{n}(gf)
= H_{n}(g)H_{n}(f) for the inclusions f:A_{4}→S_{4},
g:S_{4}→S_{5} and n=2. 

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 

