1. An Aside from Number Theory: The Euler Phi-function (9/20/13) Definition. The Euler Phi-function of a positive integer n, denoted  (n) is the number of elements in {1, 2,..., n -1} which are relatively prime to n. Here’s how to compute  (n): Factor n into its unique prime power factorization (possible by the Fundamental Theorem of Arithmetic). For each factor of the form p k (where p is prime),  (p k ) = p k – p k-1. Multiply the individual answers together to get the final  (n). What is  (17)? What is  (20)? What is  (108)? What is  (120)?

2. More on Subgroups We saw last time that a rich source of subgroups inside a group G is its cyclic subgroups. But not all subgroups are cyclic. Q is a subgroup of R. Is Q cyclic? Find a non-cyclic proper subgroup of D 4. Find another. How many are there? Find a non-cyclic proper subgroup of U(20). Of course, if G itself is non-cyclic, there’s a non-cyclic subgroup right there.

3. The Center of a Group Definition: Let G be a group and let Z(G) = {a  G | a x = x a for all x  G}. Z(G) is called the center of G. Note: This notation is Z, not Z. Both come from German. Z is for Zentrum, which means center. Z is for zahlen, which means to count. This idea is only of interest in non-abelian groups. (Why?) What is the center of D 4 ? Of D 5 ? Of D n (two cases!)? What is the center of GL(2, R)? Of SL(2, R)? Theorem. The center of any group G is a subgroup of G. Proof of theorem?

4. Assignment for Monday Continuing in Chapter 3, please do Exercises 11, 15 (Can you generalize this result? What was true of 7 and 3 that would be true of other number pairs?), 18, 19, 21, 22, 25, 26, 27, 28, 30 on pages 69-70.