  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.
