Number of Generators of a Cyclic Group of Order 12

A cyclic group is a group that can be generated by a single element. The order of a cyclic group is the number of elements it contains. In this case, we are considering a cyclic group of order 12.

Definition of a Generator

A generator of a cyclic group is an element that, when raised to different powers, produces all the elements of the group. In other words, a generator is an element whose powers generate all the other elements of the group.

Properties of a Cyclic Group

- A cyclic group of order n has exactly φ(n) generators, where φ is Euler's totient function.
- Euler's totient function φ(n) gives the count of positive integers less than n that are coprime to n.
- For a prime number p, a cyclic group of order p has p-1 generators.

Determining the Number of Generators

In this case, we have a cyclic group of order 12. To determine the number of generators, we need to find the count of positive integers less than 12 that are coprime to 12.

Calculating Euler's Totient Function

To calculate φ(12), we need to find the count of positive integers less than 12 that are coprime to 12.

The positive integers less than 12 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

Out of these, the coprime numbers to 12 (i.e., numbers that have no common factors with 12) are: 1, 5, 7, 11.

Therefore, φ(12) = 4.

Number of Generators

Since the number of generators of a cyclic group of order n is given by φ(n), we have φ(12) = 4.

Hence, the number of generators of a cyclic group of order 12 is 4.