Are generators of finite cyclic groups unique?

Question

Are generators of finite cyclic groups unique? Can someone explain to me why they are unique or why they are not?

Answer 1

No.

Hint: consider $\mathbb Z_7$; you can check by hand that it has more than one generator.

It is also easy to check that for any $[m]$ relatively prime to $n$, $[m]$ generates $Z_n$.

Answer 2

Hint: Consider the cyclic group of size $3$, $$ C_3 = \{e, a, a^2\}, $$ where $a^3 = e$.

Which of the elements are generators? You can try all three of them. Is $e$ a generator? is $a$ a generator? And is $a^2$ a generator?

Answer 3

If $g$ is a generator of the cyclic group $G$ of order $n$ (in multiplicative notation), every element of $G$ can be written in the form $g^k$, where $0\le k <o(g)$.

Furthermore, the order of such an element is $$o(g^k)=\frac{o(g)}{\gcd(o(g),k)}=\frac n{\gcd(n,k)},$$ hence an element $g^k$ is a generator of $g$ if and only if $k$ is coprime to $n$. There are $\varphi(n)$ such elements.