The reduced Gassner representation restricted to a normal free subgroup of the pure braid group (Q1364589)

As it was proved by \textit{S. Lipschutz} [Arch. Math. 12, 7-12 (1961; Zbl 0134.26001)], the reduced Gassner representation of the pure braid group \(P_n\) is faithful if and only if its restriction to a certain subgroup \(U_n\) is faithful. Let \(\rho_n\) be the restricted reduced Gassner representation, \(\rho_n\colon U_n\to\text{GL}_{n-1}(\mathbb{Z}[y^{\pm 1}_1,\dots,y^{\pm 1}_n])\). Let \(\tau\) be the homomorphism \(\mathbb{Z}[y^{\pm 1}_1,\dots,y^{\pm 1}_n]\to\mathbb{Z}[y^{\pm 1}_1,\dots,y^{\pm 1}_{n-1}]\) defined by \(\tau(y_i)= y_i\) (\(i=1,\dots,n-1\)), \(\tau(y_n)=1\), and let \(\mu\) denote the group homomorphism of the respective \(\text{GL}_{n-1}\)'s it induces. The author proves that the composition \(\mu\circ\rho_n\) is up to isomorphism the Magnus representation of \(U_n/U_n''\). Then, as a corollary, Lipschutz's result follows easily, that is, in general, the kernel of the reduced Gassner representation restricted to \(U_n\) is a subgroup of \(U_n''\).