It is suggested that while the theory of compact groups leads to an elegant mathematical formalism for calculating the properties of many-electron systems, it does not lead to a physically significant interpretation of these properties. The desirability of applying non-compact groups to many-electron systems is discussed. As a preliminary study the tensor operator algebra of SO(4) is developed using the known SO(4) vector coupling coefficients which in turn are used to study the canonical chain SO(5) ⊃; SO(4) ⊃; SO(3) ⊃; SO(2) which finds a physical realization in the dynamical symmetry of the bound states of the hydrogen atom. These results are then extended to the representations of the non-compact de Sitter group SO(4, 1) constructed in the canonical basis SO(4, 1) ⊃; SO(4) ⊃; SO(3) ⊃; SO(2). The possibility of applying the theory of non-compact groups to many-electron atoms is then considered