# 2-Divisible groups over Z by Abrashkin V.A.

By Abrashkin V.A.

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12 Circle inversion. Besides the circle inversion Ri, by composing it with isometries maintaining invariant the circle line c of the inversion circle c(0, r)—with a reflection with reflection line containing the circle center O or with a rotation with the rotation center O, we have two more conformal transformations: (i) inversive reflection Zj = RjR = RRi, the involutional transformation, the commutative composition of a reflection and a circle inversion; (ii) inversive rotation Si = SRi = RiS, the commutative composition of a rotation and a circle inversion (Fig.

In the same way, it is de- 20 Introduction fined an external automorphism of the rotation group of square H, given by presentation {S} SA = E : SR = S3, (S2)R = S2, (S3)R = S, where the reflection line of reflection R contains the center of four-fold rotation S. Hence, external automorphisms are very efficient tool for extending symmetry groups. Since the product of direct transformations is a direct transformation, and the inverse of a direct transformation is a direct transformation, each group of transformations G, which contains at least one indirect transformation has a subgroup of the index 2, denoted by G + , which consists of direct transformations of the group G.

That on conformal symmetry) demand the introduction of new symbols. Since only lately there have been attempts to make consistent and uniform the symbols of symmetry groups, positive results are mainly achieved with the symmetry groups of ornaments G-i (International symbols). In the other cases, a great number of authors, with their original results introduced together new or modified symbols. Therefore, it is unavoidable to accept the compromise solution and quote several alternative kinds of symbols.