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The lower central series of a group G is defined by setting G1=G and Gn+1=[Gn,G] . Each quotient Ln(G)=Gn/Gn+1 is an abelian group. We can form the direct sum L(G) = L1(G) + L2(G) + ... of all the quotients Ln(G). Commutation in G induces bilinear functions Lm(G)×Ln(G) → Lm+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 Ln(G) is free abelian then we take the ground ring of L(G) to be the ring of integers. If each quotient Ln(G) is an elementary abelian p-group then we take the ground ring of L(G) to be the field of p-elements. 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 H4(G,Z) and the integral Lie homology H4(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), 49-82].
Theorem Let G be the free nilpotent group of class 2 on a finite number of generators. Then the integral group homology Hi(G,Z) is isomorphic to the integral Lie algebra homology Hi(L(G),Z) in each dimension i.
The following commands exhibit a torsion free nilpotent group G of class 2 for which each quotient Ln(G) is free abelian and for which H3(G,Z) is not isomorphic to H3(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 ]
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