Identifications in modular group algebras

Let G be a group, S a subgroup of G, and image a field of characteristic p. We denote the augmentation ideal of the group algebra image by ω(G). The Zassenhaus–Jennings–Lazard series of G is defined by Dn(G)=G∩(1+ωn(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with image is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the Dn(G). We then extend a theorem of Jennings that provides a basis for the quotient ωn(G)/ωn+1(G) in terms of a basis of the restricted Lie algebra associated with the Dn(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωn(S))=Dn+1(S) and 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.