The intimate relation of symmetry properties describable by groups of motions and the consequent conservation laws realizable for a particle in classical mechanics and in the mechanics of restricted and general theories of relativity are discussed in detail using elementary results of the theory of continuous groups. In addition, for the case of special relativistic mechanics the finite form of the groups of motions, underlying all of the possible constants of the motion of the form of first integrals linear in the momenta, are constructed and shown to be the inhomogeneous (i.e., including displacements) proper Lorentz transformations.

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