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Unformatted text preview: 90 2. BASIC THEORY OF GROUPS which involve products of no more that n 1 elements at a time, are de- fined. Moreover, we have p k D p k C 1 for 1 k n 2 , since p k D .a 1 a k /.a k C 1 a n / D .a 1 a k /.a k C 1 .a k C 2 a n // D ..a 1 a k /a k C 1 /.a k C 2 a n / D .a 1 a k C 1 /.a k C 1 a n / D p k C 1 : Thus all the products p k are equal, and we can define the product of n elements satisfying (a)-(c) by a 1 a n D a 1 .a 2 a n /: n Exercises 2.1 2.1.1. Determine the symmetry group of a nonsquare rhombus. That is, de- scribe all the symmetries, find the size of the group, and determine whether it is isomorphic to a known group of the same size. If it is an entirely new group, determine its multiplication table. 2.1.2. Consider a square with one pair of opposite vertices painted red and the other pair of vertices painted blue. Determine the symmetry group of the painted square. That is, describe all the (color-preserving) symme- tries, find the size of the group and determine whether it is isomorphic to...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08