


The
lower central series of a group G is defined by setting G_{1}=G
and G_{n+1}=[G_{n},G] . Each quotient L_{n}(G)=G_{n}/G_{n+1}
is an abelian group. We can form the direct sum L(G) = L_{1}(G)
+ L_{2}(G) + ... of all the quotients L_{n}(G).
Commutation in G induces bilinear functions L_{m}(G)×L_{n}(G)
→ L_{m+n}(G) which provide L(G) with a bilinear bracket
operation [ , ]:L(G)×L(G) → L(G). This operation satisfies the
Jacobi identity, and L(G) can thus be considered as a Lie algebra. If
each quotient L_{n}(G) is free abelian then we take the ground
ring of L(G) to be the ring of integers. If each quotient L_{n}(G)
is an elementary abelian pgroup then we take the ground ring of L(G)
to be the field of pelements. We shall not consider groups for which
L(G) is neither free nor elementary abelian. The following command creates L(G) for G equal to the free nilpotent group of class 2 on four generators. 

gap>
F:=FreeGroup(4);; G:=NilpotentQuotient(F,2);; gap> LG:=LowerCentralSeriesLieAlgebra(G); <Lie algebra of dimension 10 over Integers> 

The
following additional commands calculate the integral group homology H_{4}(G,Z)
and the integral Lie homology H_{4}(L(G),Z) . 

gap>
GroupHomology(G,4); [ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> LieAlgebraHomology(LG,4); [ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 

We
see that the group homology and Lie homology are equal. This
illustrates the following theorem in [Yu. V. Kuz'min and Yu. S.
Semenov, "On the homology of a free nilpotent group of class 2", Mat.
Sb. 189, no. 4 (1998), 4982].
The following commands exhibit a torsion free nilpotent group G of class 2 for which each quotient L_{n}(G) is free abelian and for which H_{3}(G,Z) is not isomorphic to H_{3}(L(G),Z). This suggests that the "free nilpotent" hypothesis in the theorem can not be weakened. 

gap>
G:=HeisenbergPcpGroup(3);; LG:=LowerCentralSeriesLieAlgebra(G);; gap> GroupHomology(G,3); [ 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> LieAlgebraHomology(LG,3); [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 

However,
extensive experimentation on free nilpotent groups of class >2
suggest that it might be possible to drop the "class 2" hypothesis of
the theorem. For example: 

gap>
F:=FreeGroup(3);; G:=NilpotentQuotient(F,3);; gap> LG:=LowerCentralSeriesLieAlgebra(G); gap> GroupHomology(G,4); #This command takes several hours!! [ 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> LieAlgebraHomology(LG,4); [ 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] 

