We reconsider the definitions of both material symmetries and physical symmetries which are described in terms of point groups, i.e. subgroups of the full orthogonal group, because these two concepts are often confused, and the classical descriptions of physical symmetry for inelastic behaviour of materials are impracticable. All two- and three- dimensional point groups are classified into two types: compact and non- compact. The reality of every compact point group in the description of a material or a physical symmetry is justified in four aspects, that is: (i) point groups characterized by a finite set of tensors, (ii) Hilbert’s theorem for integrity bases, (iii) correlation between integrity bases and function bases (generalization of Wineman and Pipkin’s theorem), and (iv) physical reality.