The isometric symmetry groups in the plane E2 can be classified according to the spaces invariant with respect to the action of transformations of the groups in question. Bohm symbols have been used to denote the corresponding categories of symmetry groups (J. Bohm, K. Dornberger-Schiff, 1966). In the symbol Gnst¼, the first subscript n represents the maximal dimension of space in which the transformations of the symmetry group act, while the following subscripts st¼ represent the maximal dimensions of subspaces that are invariant with respect to the action of transformations of the symmetry group and that are properly included in each other. These symbols represent also the definitions of the corresponding categories of isometric symmetry groups in E2: the symmetry groups of finite friezes G210, rosettes G20, friezes G21, and ornaments G2. In line with the relation G210 Ì G20, and to simplify things, the category G210 will not be discussed individually but within the category G20.
Antisymmetry and colored symmetry, the extensions of the theory of symmetry, will be used only for a more detailed analysis of the symmetry groups in E2.