That is, for any n>0 and group homomorphism f : G → G' there is an induced
homomorphism Hn(f) : Hn(G,Z) → Hn(G',Z)
gap> f:=GroupHomomorphismByFunction(P,S_5, x->x);;
above computation illustrates a general result.
We denote by Hn(G,Z)(p) the p-part of Hn(G,Z). This result follows from a property of the transfer homomorphism
Tr(G,K) : Hn(G,Z) → Hn(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 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
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(P,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:
gap> gensP:=[gens,gens,gens];; #This list generates a Sylow 2-subgroup of M_23.
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
..., 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 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 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.)
[ 2, 2, 2, 2, 2, 2, 2 ]
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.