On modular relation modules and residual nilpotence

Abstract

Let F be a non-cyclic free group, let R d F, and let G = FIR. In [3], Gruenberg showed that if G is finite, then FIR’ is residually nilpotent if and only if G is a finite p-group. In that paper, the connection between this question and the intersection of the powers of the augmentation ideal 0 of the integral group ring ZG, was brought out. It was natural then for the same question to be taken up for infinite G, and this was done by several authors, culminating in the paper of Lichtman [14]. There, completely general conditions on G were given, equivalent to the residual nilpotence of F/R’. In [7], this work was extended to give necessary and sufficient conditions for F/S to be residually nilpotent, where S is a fully invariant subgroup of R such that R/S is a non-trivial torsion-free nilpotent group. For more details on these matters, see [g]. It seems natural to investigate the same question when R/S is not torsion-free, and in particular when it is a relatively free nilpotent p-group, and this is one of the themes of this paper. Much remains to be said here, but we shall prove the following result. If H is a group and p a prime, we write D,,(H, p”‘) for the nth dimension subgroup of H over the ring of integers modulo p’“. This is clearly a fully invariant subgroup of H.