Abstract Algebra Theory and Applications - Computer Science ...

180 CHAPTER 10 MATRIX GROUPS AND SYMMETRYConsequently,〈f(x), f(y)〉 = 〈x, y〉.Now let e 1 and e 2 be (1, 0) t and (0, 1) t , respectively. Ifthenx = (x 1 , x 2 ) = x 1 e 1 + x 2 e 2 ,f(x) = 〈f(x), f(e 1 )〉f(e 1 ) + 〈f(x), f(e 2 )〉f(e 2 ) = x 1 f(e 1 ) + x 2 f(e 2 ).The linearity of f easily follows.For any arbitrary isometry, f, T x f will fix the origin for some vectorx in R 2 ; hence, T x f(y) = Ay for some matrix A ∈ O(2). Consequently,f(y) = Ay + x. Given the isometriestheir composition isf(y) = Ay + x 1g(y) = By + x 2 ,f(g(y)) = f(By + x 2 ) = ABy + Ax 2 + x 1 .This last computation allows us to identify the group of isometries on R 2with E(2).□Theorem 10.4 The group of isometries on R 2E(2).is the Euclidean group,A symmetry group in R n is a subgroup of the group of isometries onR n that fixes a set of points X ⊂ R 2 . It is important to realize that thesymmetry group of X depends both on R n and on X. For example, thesymmetry group of the origin in R 1 is Z 2 , but the symmetry group of theorigin in R 2 is O(2).Theorem 10.5 The only finite symmetry groups in R 2 are Z n and D n .Proof. Any finite symmetry group G in R 2 must be a finite subgroup ofO(2); otherwise, G would have an element in E(2) of the form (A, x), wherex ≠ 0. Such an element must have infinite order.By Example 6, elements in O(2) are either rotations of the form( )cos θ − sin θR θ =sin θ cos θ