Abstract Algebra Theory and Applications - Computer Science ...

188 CHAPTER 10 MATRIX GROUPS AND SYMMETRY(b) The quaternion group, Q 8 , cannot be written as a semidirect product.(c) E(2) is the semidirect product of O(2) by H, where H consists of alltranslations in R 2 .16. Determine which of the 17 wallpaper groups preserves the symmetry of thepattern in Figure 10.5.Figure 10.11.17. Determine which of the 17 wallpaper groups preserves the symmetry of thepattern in Figure 10.11.18. Find the rotation group of a dodecahedron.19. For each of the 17 wallpaper groups, draw a wallpaper pattern having thatgroup as a symmetry group.References and Suggested Readings[1] Coxeter, H. M. and Moser, W. O. J. Generators and Relations for DiscreteGroups, 3rd ed. Springer-Verlag, New York, 1972.[2] Grove, L. C. and Benson, C. T. Finite Reflection Groups. 2nd ed. Springer-Verlag, New York, 1985.[3] Hiller, H. “Crystallography and Cohomology of Groups,” American MathematicalMonthly 93 (1986), 765–79.[4] Lockwood, E. H. and Macmillan, R. H. Geometric Symmetry. CambridgeUniversity Press, Cambridge, 1978.[5] Mackiw, G. Applications of Abstract Algebra. Wiley, New York, 1985.[6] Martin, G. Transformation Groups: An Introduction to Symmetry. Springer-Verlag, New York, 1982.[7] Milnor, J. “Hilbert’s Problem 18: On Crystallographic Groups, FundamentalDomains, and Sphere Packing,” Proceedings of Symposia in Pure Mathematics18, American Mathematical Society, 1976.[8] Phillips, F. C. An Introduction to Crystallography. 4th ed. Wiley, New York,1971.