Band and Curie limit symmetry groups

The paper discusses the contradiction between the 7 band (on a cylinder with infinite radius) symmetry groups and the 5 uniaxial Curie limit symmetry groups. Logical difficulties in understanding the symmetry axis ∞ as a true crystallographic one are shown. The formula n → ∞ is proposed to be understood as if the order n of the axis becomes as large as desired, but retains the properties of a natural number (even, odd etc.). In this way, the true inversion axes of symmetry and one-to-one correspondence of bands and limit groups are restored. Such an analysis may be useful in teaching a university course in crystallography.