  We consider renormalization groups of transformations composed of a Gaussian
convolution and a field dilatation. As an example, we consider perturbations of
a single component real Euclidean free field $\phi$ with covariance
$(-\bigtriangleup)^{-1+\frac{\epsilon}{2}}$. We show that the renormalization
group admits two equivalent formulations called massless picture and massive
picture respectively. We then show in the massive picture that the
renormalization group has a symmetry. The symmetry consists of global scale
transformations composed with certain Gaussian convolutions. We translate the
symmetry back to the massless picture. The relation between the symmetry and
the notion of an anomalous dimension is briefly discussed.
