On Fox Quotients of Arbitrary Group Algebras

For a group G and N-series G of G let In R,G(G), n ≥ 0, denote the filtration of the group algebra R(G) induced by G , and IR(G) its augmentation ideal. For subgroups H of G , left ideals J of R(H) and right H -submodules M of IZZ(G) the quotients IR(G)J/MJ are studied by homological methods, notably for M = IZZ(G)IZZ(H), IZZ(H)IZZ(G) + IZZ([H,G])ZZ(G) and ZZ(G)IZZ(N) + In ZZ,G(G) with N CG where the group IR(G)J/MJ is completely determined for n = 2. The groups In−1 ZZ,G (G)IZZ(H)/I n ZZ,G(G)IZZ(H) are studied and explicitly computed for n ≤ 3 in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups.