The de Sitter and anti-de Sitter spacetimes are transitive under a mixture of translations and proper conformal transformations. The relative importance of each one of these transformations is determined by the value of the cosmological constant $\Lambda$. For a vanishing $\Lambda$, both de Sitter groups are reduced to the Poincar\'e group, and both de Sitter spaces become the Minkowski spacetime, which is transitive under ordinary translations. For an infinite cosmological constant, the resulting spacetime is a singular, four-dimensional cone-space, transitive under {\it proper} conformal transformations. The geometric properties of this cone-space are studied, and its possible relation with the initial conditions of a big bang universe discussed.