The authors correlate the symmetry group of the continuous transformations of the Toda lattice to that of the Korteweg-de Vries equation. They show how, by taking into account the continuous limit of the Toda lattice the four-parameter symmetry group of the Toda lattice is contained in that of the KdV equation. By an inverse process, discretization of the symmetry group of the KdV equation, they find a discrete element of the symmetry group of the Toda lattice, which gives, by symmetry reduction, its soliton solution.