A note on free presentations and residual nilpotence (Q793152)

A group H is said to be discriminated by nilpotent groups of prime power exponent if, given any finite set \(h_ 1,h_ 2,...,h_ n\) of non- trivial elements of H, there exists a nilpotent group K of (finite) prime power exponent and a homomorphism \(\phi:H\to K\) such that \(\phi(h_ i)\neq 1\) for \(i=1,2,...,n.\) Let \({\mathfrak D}\) denote the class of all groups that are either residually torsion-free nilpotent or discriminated by nilpotent groups of finite prime power exponent. The main result of this paper is the following Theorem. Let F be a non-cyclic free group, \(R\triangleleft F\), and S be a fully invariant subgroup of R such that R/S is non-trivial, torsion-free and nilpotent. Then the following conditions are equivalent: (a) F/S is residually nilpotent, (b) \(F/S\in {\mathfrak D},\) (c) \(F/R\in {\mathfrak D}.\)