The kinetic and thermodynamic stabilities of the group 13 hydrides EH3 (E = B, Al, Ga, In, Tl, E113) are investigated by relativistic density functional and wave function based theories. The unimolecular decomposition of EH3 --> EH + H-2 becomes energetically more favorable going down the Group 13 elements, with the H-2-abstraction of InH3, TlH3, and (E113)H-3 (E113: element with nuclear charge 113) being exothermic. In accordance with the Hammond-Leffler postulate, the activation barrier for the dissociation process decreases accordingly going down the group 13 elements in the periodic table shifting to an early transition state, with activation energies ranging from 88.4 kcal/mol for BH3 to 41.3 kcal/mol for TlH3 and only 21.6 kcal/mol for (E113)H-3 at the scalar relativistic coupled cluster level of theory. For both TlH3 and (E113)H-3 we investigated spin-orbit effects using Dirac-Hartree-Fock and second-order Moller-Plesset theory to account for electron correlation. For (E113)H-3 spin-orbit coupling results in a chemically inert closed 7p(1/2)-shell, thus reducing the stability of the higher oxidation state even further. We also investigated the known organothallium compound Tl(CH3)(3), which is thermodynamically unstable similar to TlH3, but kinetically very stable with an activation barrier of 57.1 kcal/mol.