Question

If G is a cyclic group of order 12, then the order of Aut(G) is:

• A. 1
• B. 5
• C. 4
• D. 77

Hint:

Remembering the relationship \( |\text{Aut}(\mathbb{Z}_n)| = \phi(n) \) is a major shortcut for this type of problem. An automorphism of \(\mathbb{Z}_n\) is determined by where it sends the generator 1. It must send 1 to another generator, and the generators of \(\mathbb{Z}_n\) are precisely the integers \(k\) such that \(1 \le k<n\) and gcd(k,n)=1. The number of such integers is \( \phi(n) \).

Answer 1

The Correct Option is C

Step 1: Problem Definition:
The task is to determine the order of the automorphism group of a cyclic group with order 12. An automorphism is a self-isomorphism, and the group of all automorphisms, denoted Aut(G), forms a group under composition.

Step 2: Core Concept:
For a cyclic group of order \(n\), \( G \cong \mathbb{Z}_n \), the automorphism group Aut(G) is isomorphic to the group of units modulo \(n\), \( U(n) \). The order of this group is given by Euler's totient function, \( \phi(n) \). Therefore: \[ |\text{Aut}(\mathbb{Z}_n)| = |U(n)| = \phi(n) \] The goal is to calculate \( \phi(12) \).

Step 3: Solution:
Euler's totient function, \( \phi(n) \), counts the integers relatively prime to \(n\) within the range 1 to \(n\). We can compute \( \phi(12) \) in two ways: 1. By enumeration: List integers from 1 to 11. Identify those coprime with 12: {1, 5, 7, 11}. Thus, \( \phi(12) = 4 \). 2. Using the formula: Factorize 12: \( 12 = 2^2 \times 3^1 \). Apply the formula: \( \phi(n) = n \prod_{p|n, p \text{ is prime}} (1 - \frac{1}{p}) \). \[ \phi(12) = 12 \left(1 - \frac{1}{2}\right) \left(1 - \frac{1}{3}\right) = 12 \times \frac{1}{2} \times \frac{2}{3} = 4 \]
Step 4: Answer:
The order of Aut(G) is 4.