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By Ol'shanskii A. Y.

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Example text

In this case H is not a normal subgroup of G. The order of the group P/v is JViVi, where [G : H] = N, [H : Hi] = Ni and quotient group G/Hi = PN. To denote such colored symmetry groups, the symbols G/H/Hi are used (Fig. 18b). ). Taking into consideration the abundance of unused possibilities, although being in use for centuries, colored symmetry can be a very inspiring field for creative artistic work and scientific research. We find proof of this in the works of Escher (Escher, 1971a, b; Schattschneider, 1990).

All symmetry groups will be subgroups of groups generated by reflections. In the case of conformal symmetry groups, along with reflections, circle inversions have the analogous function. For example, the symmetry group of square is the group generated by reflections, with the fundamental region of the fixed shape (Fig. 3a). The results of composition of plane isometries are different categories of groups of isometries of the space E2, represented by Bohm symbols as: G210—symmetry groups of finite friezes, G20—symmetry groups of rosettes, G21—symmetry groups of friezes and G2—symmetry groups of ornaments.

9). The existence of indirect symmetries of a figure / implies the absence of enantiomorphism. Fig. 9 "Left" and "right" rosette with the symmetry group C4, consisting of direct symmetries. Since reflections have a role of elementary isometric transformations, while all other isometries are their finite compositions, of special interest will be symmetry groups generated by reflections—groups, a set of generators of which consists exclusively of reflections. Since every reflection keeps invariant each point of the reflection line, the fundamental region of these groups will possess a fixed shape, will not allow variations and will have Classification of Symmetry Transformations and Groups 21 rectilinear edges.

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A. I. Maltsevs problem on operations on groups by Ol'shanskii A. Y.

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