Cayley Tables (9/9)

1. 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: Z4 , U(12) , D3, D4

2. 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 D4 , the symmetries of a square.

3. Dihedral Groups • One class of symmetry groups are the dihedral groups Dn, 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 Dnhave, and why? • Classify the elements into different types. Note that the cases n even and n odd are somewhat different. Look at some pictures!

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