The symmetry inherent in the periodic system of elements is discussed on the basis of the model one-electron potential in atoms. This potential approximates well that of the Thomas-Fermi theory and reproduces the well known n+l building-up rule describing the configurations of the ground-state neutral atoms. The modifications of the model potential accounting for the Coulomb tail allows us to analyse the (n+l) grouping in the Rydberg spectra of atoms. This grouping is shown to be closely connected with the n+l building-up rule for the ground states. The dynamic group of the periodic system (or of the atomic potential) is proposed. The specific doubling of the periods (as compared with the hydrogen type spectrum) is described by means of a group formally analogous to the isospin group. The operator of the 'isospin' projection on the chosen axis in the 'isospace' is found explicitly as an inversion operation in some sphere. The states classification according to the related quantum number allows us to give the group-theoretical interpretation of the 'secondary periodicity' effect in the periodic system.