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.

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 Z15, are 5 and 20(2) the same?

How about 5 and 25(2)?

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 ak generates a if and only if GCD(k, n) = 1.

• Example: Does 14 generate Z91? How about 15?

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?

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.

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.