Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group
G5 is suggested as the symmetry group of dynamics.
G5 contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of
G5 by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.
In several recent papers on entanglement in relativistic quantum systems and relativistic Bell's inequalities, relativistic Bell-type two-particle states have been constructed in analogy to nonrelativistic states. These constructions do not have the form suggested by relativistic invariance of the dynamics. Two relativistic formulations of Bell-type states are shown for massive particles, one using the standard Wigner spin basis and one using the helicity basis. The construction hinges on the use of Clebsch-Gordan coefficients of the Poincar\'e group to reduce the direct product of two unitary irreducible representations (UIRs) into a direct sum of UIRs.