On modular relation modules and residual nilpotence
Let \(F\) be a free non-cyclic group with a normal subgroup \(R\) and \(G=F/R\). Denote by \(S\) the \(n\)th dimension subgroup \(D_ n(R,p^ m)\), where \(p\) is a prime and \(n\), \(m\) are some integers. The following are equivalent: (i) \(F/S\) is residually nilpotent; (ii) \(F/S\) is residually a nilpotent \(p\)-group of finite exponent; (iii) \(G\) is residually a nilpotent \(p\)-group of finite exponent; (iv) the intersection of the powers of the augmentation ideal of \(G\) over \(\mathbb{F}_ p\) is zero. Suppose now that \(M\) is a verbal subgroup in \(R\) corresponding to a set of outer commutator words, \(M\subseteq \gamma_ c(R)\), where \(\gamma_ c(R)\) is the \(c\)th member of the lower central series in \(R\). Put \(\overline {S}=M/[M\cap \gamma_{c+1}(R)\gamma_ c(R)^ p]\). Suppose that \(\overline{S}\neq 0\) and \(G=F/R\) is infinite. Then \(\overline {S}\) is a faithful \(\mathbb{F}_ p G\)-module.
scientific article; zbMATH DE number 427659