Question

Find the number of automorphisms of the cyclic group $\mathbb{Z}_n$ for $n = 30$.

Hint:

Number of automorphisms of $\mathbb{Z}_n$ equals $\varphi(n)$ because automorphisms correspond to choosing generators.

Answer 1

Correct Answer: 8

Step 1: Use the standard result.
For a cyclic group $\mathbb{Z}_n$, the number of automorphisms is:
\[ |\text{Aut}(\mathbb{Z}_n)| = \varphi(n), \] where $\varphi(n)$ is Euler’s totient function.
Step 2: Prime factorization of 30.
\[ 30 = 2 \times 3 \times 5. \]
Step 3: Apply Euler’s Totient Formula.
For \[ n = p_1 p_2 p_3, \] \[ \varphi(n) = n \left(1 - \frac{1}{p_1}\right) \left(1 - \frac{1}{p_2}\right) \left(1 - \frac{1}{p_3}\right). \]
Thus, \[ \varphi(30) = 30\left(1-\frac{1}{2}\right) \left(1-\frac{1}{3}\right) \left(1-\frac{1}{5}\right). \]
\[ = 30 \times \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5}. \]
\[ = 15 \times \frac{2}{3} \times \frac{4}{5}. \]
\[ = 10 \times \frac{4}{5}. \]
\[ = 8. \]
Final Answer:
\[ \boxed{8}. \]