Cyclic Groups (9/25)

1. Cyclic Groups (9/25) • Definition. A group G is called cyclicif 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 jif and only if i = j . If a has finite order n, thena i= a jif 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 ak= e, then n divides k. • Example. In Z15, are 5 and 20(2) the same? How about 5 and 25(2)?

3. Generators and Smallest Generators • Theorem. Suppose |a| = n. Then ak = aGCD(n,k) and the order of ak = n /GCD(n,k). • Example (Abstract group): Suppose |a| = 20. What generator of a12 has the smallest exponent? What is the order of this subgroup? • Example (Additive group): What is the smallest generator of 9 in Z15? 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 akgenerates a if and only if GCD(k, n) = 1. • Example: Does 14 generate Z91? 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 an/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 Z12. • 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 Z91 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.