  We will say that a group G possesses the Magnus property if for any two
elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}.
We prove that some one-relator groups, including the fundamental groups of
closed nonorientable surfaces of genus g>3 possess this property. The analogous
result for orientable surfaces of any finite genus was obtained by the first
author [Geometric methods in group theory, Contemp. Math, 372 (2005) 59-69].
