Cayley Tables (9/9) If a group G is finite, we can (theoretically at least) write down the whole operation table, or Cayley table of G. We always put the elements in the same order down the left-hand side and across the top, and we always list the identity element first. If the group is abelian, the table will be symmetric about the main diagonal. If the group is non-abelian, we assume that the row a, column b entry is a b. Note then that for groups in which the operation is composition, that means first do b, then do a ! Examples: Z 4, U(12), D 3, D 4

Symmetry Groups of Objects The rigid motions of any object which return it to an identical view are called the symmetries of that object. Because the composition of two symmetries of a given object produces a symmetry (so, we have closure), because there is always the symmetry which does nothing (identity), and because any symmetry can simply be reversed (inverses), the set of all symmetries form a group, called the symmetric group of the object. Though these can apply to three dimensional objects, we shall stick (for now) with looking at two dimensional objects. Our first example has been D 4, the symmetries of a square.

Dihedral Groups One class of symmetry groups are the dihedral groups D n, the symmetries of a regular n-gon (i.e., an n-sided polygon in which all sides and angles are the same. How many elements does D n have, and why? Classify the elements into different types. Note that the cases n even and n odd are somewhat different. Look at some pictures!

Assignment for Wednesday Finish absorbing Chapter 1. On pages 38-39, do Exercises 11-15, 17, 21, 23, 24.