1. Cyclic Groups (9/25) Definition. A group G is called cyclic if there exists an element a in G such that G =  a . That is, every element of G can be written as a power of a (or, if the particular group is written additively, can be written as a multiple of a). Give a simple example an infinite cyclic group. In a sense we’ll make precise later, this example is really the infinite cyclic group. That is, the basic arithmetic in any infinite cyclic group is just the same as this arithmetic. Give a simple example of a finite cyclic group of order n. Same remark as before. This is really the finite cyclic group of order n. Note that all cyclic groups are abelian.

2. When is a i = a j ? Theorem. Let a  G. If the order of a is infinite, then a i = a j if and only if i = j. If a has finite order n, then a i = a j if and only if n divides i – j. Remark. The proof uses the so-called division algorithm from basic number theory: If n is a positive integer and k is any integer, then there exist integers q (the “quotient”) and r (the “remainder”) such that k = q(n) + r with 0  r < n. Corollary. For any a in G, |a| = |  a  |. Corollary. For any a in G of order n, if a k = e, then n divides k. Example. In Z 15, are 5 and 20(2) the same? How about 5 and 25(2)?

3. Generators and Smallest Generators Theorem. Suppose |a| = n. Then  a k  =  a GCD(n,k)  and the order of a k = n / GCD(n,k). Example (Abstract group): Suppose |a| = 20. What generator of  a 12  has the smallest exponent? What is the order of this subgroup? Example (Additive group): What is the smallest generator of  9  in Z 15 ? What is the order of this subgroup? Corollary. In a cyclic finite group, the order of any element divides the order of the group. Corollary. If |a| = n, then a k generates  a  if and only if GCD(k, n) = 1. Example: Does 14 generate Z 91 ? How about 15?

4. Fundamental Theorem of Cyclic Groups Theorem. Every subgroup of a cyclic group is cyclic. If |  a  | = n, then the order of every subgroup of  a  divides n, and for every divisor d of n, there exists exactly one subgroup of order d, namely  a n/d . This says that cyclic groups have a very simple and predictable structure. We should think of them as the simplest groups there are. Example: Write all the subgroups of Z 12. Question: Is it true in general that the orders of subgroups of a finite group must divide the order of the group? Question: Is it true that if d divides n, the order of an arbitrary group G, then there must exist a subgroup of order d? If one exists, is it unique?

5. Finally, how many generators? Theorem. If |  a  | = n, then  a  has  (n) generators. Example. How many generators does Z 91 have? Example. U(43) is cyclic. How many generators does it have? Example. In fact, U(p) is cyclic for all primes p. Hence U(p) always has  (p – 1) generators.

6. Assignment for Friday Study the slides please. Except for the proof of Theorem 4.1, read Chapter 4 “lightly” if you wish. On page 87, do Exercises 1, 2, 3, 7, 8, 9, 10,11.