Form invariant transformations between n-- and m-- dimensional flat Friedmann--Robertson--Walker cosmologies

We illustrate how the group of symmetry transformations, which preserve the form of the n--dimensional flat Friedmann--Robertson--Walker cosmologies satisfying Einstein equations, acts in any dimension. This group relates the energy density and the isotropic pressure of the cosmic fluid to the expansion rate. The freedom associated with the dimension of the space time yields assisted inflation even when the energy density of the fluid is a dimensional invariant and enriches the set of duality transformations leading to phantom cosmologies.