  In this paper the dynamical noninvariance group SO(4,2) for a hydrogen-like
atom is derived through two different approaches. The first one is by an
established traditional ascent process starting from the symmetry group SO(3).
This approach is presented in a mathematically oriented original way with a
special emphasis on maximally superintegrable systems, N-dimensional extension
and little groups. The second approach is by a new symmetry descent process
starting from the noninvariance dynamical group Sp(8,R) for a four-dimensional
harmonic oscillator. It is based on the little known concept of a Lie algebra
under constraints and corresponds in some sense to a symmetry breaking
mechanism. This paper ends with a brief discussion of the interest of SO(4,2)
for a new group-theoretical approach to the periodic table of chemical
elements. In this connection, a general ongoing programme based on the use of a
complete set of commuting operators is briefly described. It is believed that
the present paper could be useful not only to the atomic and molecular
community but also to people working in theoretical and mathematical physics.
