the Poincaré group Symmetry in quantum mechanics Wigner's classification Remarks Notes Freely available online references Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006). Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. References (the representation theory of SO(2,1) and SL(2, R); the second part on SO(3; 1) and SL(2, C), described in the introduction, was never published). (free access) (a general introduction for physicists) (elementary treatment for SL(2,C)) (a detailed account for physicists) (James K. Whittemore Lectures in Mathematics given at Yale University, 1967) , Chapter 9, SL(2, C) and more general Lorentz groups . Representation theory of Lie groups Special relativity Quantum mechanics