Abstract Algebra Theory and Applications - Computer Science ...

186 CHAPTER 10 MATRIX GROUPS AND SYMMETRYtheorem can be found in several of the references at the end of this chapter,including [5], [6], [10], and [11].Historical NoteSymmetry groups have intrigued mathematicians for a long time. Leonardo daVinci was probably the first person to know all of the point groups. At the InternationalCongress of Mathematicians in 1900, David Hilbert gave a now-famousaddress outlining 23 problems to guide mathematics in the twentieth century.Hilbert’s eighteenth problem asked whether or not crystallographic groups in ndimensions were always finite. In 1910, L. Bieberbach proved that crystallographicgroups are finite in every dimension. Finding out how many of these groups thereare in each dimension is another matter. In R 3 there are 230 different space groups;in R 4 there are 4783. No one has been able to compute the number of space groupsfor R 5 and beyond. It is interesting to note that the crystallographic groups werefound mathematically for R 3 before the 230 different types of crystals were actuallydiscovered in nature.Exercises1. Prove the identity〈x, y〉 = 1 [‖x + y‖ 2 − ‖x‖ 2 − ‖y‖ 2] .22. Show that O(n) is a group.3. Prove that the following matrices are orthogonal. Are any of these matricesin SO(n)?( √ √ )( √ √ )(a) 1/ 2 −1/ 21/ √ 2 1/ √ (b) 1/ 5 2/ 52−2/ √ 5 1/ √ 5(c)⎛⎝4/ √ 5 0 3/ √ 5−3/ √ 5 0 4/ √ 50 −1 0⎞⎠(d)⎛⎝1/3 2/3 −2/3−2/3 2/3 1/3−2/3 1/3 2/3⎞⎠4. Determine the symmetry group of each of the figures in Figure 10.10.5. Let x, y, and w be vectors in R n and α ∈ R. Prove each of the followingproperties of inner products.(a) 〈x, y〉 = 〈y, x〉.