Abstract Algebra Theory and Applications - Computer Science ...

184 CHAPTER 10 MATRIX GROUPS AND SYMMETRYa lattice. Space groups are, of course, infinite. Using geometric arguments,we can prove the following theorem (see [5] or [6]).Theorem 10.6 Every translation group in R 2 is isomorphic to Z × Z.SquareRectangularRhombicParallelogramHexagonalFigure 10.8. Types of lattices in R 2The point group of G is G 0 = {A : (A, b) ∈ G for some b}. In particular,G 0 must be a subgroup of O(2). Suppose that x is a vector in a latticeL with space group G, translation group H, and point group G 0 . For anyelement (A, y) in G,(A, y)(I, x)(A, y) −1 = (A, Ax + y)(A −1 , −A −1 y)= (AA −1 , −AA −1 y + Ax + y)= (I, Ax);hence, (I, Ax) is in the translation group of G. More specifically, Ax mustbe in the lattice L. It is important to note that G 0 is not usually a subgroupof the space group G; however, if T is the translation subgroup of G, thenG/T ∼ = G 0 . The proof of the following theorem can be found in [2], [5],or [6].Theorem 10.7 The point group in the wallpaper groups is isomorphic toZ n or D n , where n = 1, 2, 3, 4, 6.To answer the question of how the point groups and the translationgroups can be combined, we must look at the different types of lattices.Lattices can be classified by the structure of a single lattice cell. The possiblecell shapes are parallelogram, rectangular, square, rhombic, and hexagonal