The kinematical groups are classified; they include, besides space-time translations and spatial rotations, ``inertial transformations'' connecting different inertial frames of reference. When parity and time-reversal are required to be automorphisms of the groups, and when a weak hypothesis on causality is made, the only possible groups are found to consist of the de Sitter groups and their rotation-invariant contractions. The scheme of the contractions connecting these groups enables one to discuss their physical meaning. Beside the de Sitter, Poincaré, and Galilei groups, two types of groups are found to present some interest. The ``static group'' applies to the static models, with infinitely massive particles. The other type, halfway between the de Sitter and the Galilei groups, contains two nonrelativistic cosmological groups describing a nonrelativistic curved space-time.

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All the abstract ten-dimensional real Lie algebras that contain as a subalgebra the algebra of the three-dimensional rotation group (generators J) and decompose under the rotation group into three three-vector representation spaces (J itself, K, and P) and a scalar (generator H) are classified. In all cases, the existence of a homogeneous space of dimension 4 is shown.