Identifications in modular group algebras

Let GG be a group, SS a subgroup of GG, and FF a field of characteristic pp. We denote the augmentation ideal of the group algebra FGFG by ω(G)ω(G). The Zassenhaus–Jennings–Lazard series of GG is defined by Dn(G)=G∩(1+ωn(G))Dn(G)=G∩(1+ωn(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FGFG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the Dn(G)Dn(G). We then extend a theorem of Jennings that provides a basis for the quotient ωn(G)/ωn+1(G)ωn(G)/ωn+1(G) in terms of a basis of the restricted Lie algebra associated with the Dn(G)Dn(G). We shall use these theorems to prove the main results of this paper. For GG a finite pp-group and nn a positive integer, we prove that G∩(1+ω(G)ωn(S))=Dn+1(S)G∩(1+ω(G)ωn(S))=Dn+1(S) and G∩(1+ω2(G)ωn(S))=Dn+2(S)Dn+1(S∩D2(G))G∩(1+ω2(G)ωn(S))=Dn+2(S)Dn+1(S∩D2(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.