  Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations
(NGT) defined in terms of a wave function $\psi(x)$ do not form a group. To get
a group property one has to consider transformations that act differently on
different branches of the complex argument function and the knowledge of the
value of $\psi(x)$ is not sufficient for a well defined NGT. NGT that are well
defined in terms of $\psi(x)$ form a semigroup parametrized by a real number
$\gamma$ and a nonzero $\lambda$ which is either an integer or $-1\leq
\lambda\leq 1$. An extension of NGT to projectors and general density matrices
leads to NGT with complex $\gamma$. Both linearity of evolution and Hermiticity
of density matrices are gauge dependent properties.
