Some Properties of Elements of the Group F/[N,N]

Abstract

Let F be a free group with basis {x_j|j ∈ J}; N a normal subgroup of F. For a given element n of N we describe an elements D_l(n), where D_l: Z(F) → Z(F) (l ∈ J) are the Fox derivations of the group ring Z(F). If r₁, r₂ are an elements of F/[N,N] and, for some positive integer d, r₁^d is in the normal closure of r₂^d in F/[N,N], then r₁ is in the normal closure of r₂ in F/[N,N]. Let F/N be a soluble group; r an element of F, R the normal closure of r in F. If, for some positive integer k, r ∉ N^(k) and F/RN^(k) is torsion free then F/RN^(k+1) is torsion free.