If $G$ be a cyclic group of prime order $p$, prove that every non-identity element of $G$ is a generator of $G$.
WE know that no of generator= $\phi{(p)}=p-1$.
We have, if $a$ is a generator of $G$ then $a^k$ is a generator iff $gcd(p,k)=1$. Here as $p$ is prime, then $k=1, 2,3,\cdots, p-1$ as gcd of each of $k$ with $p$ is $1$. Therefore, generator of $G$ are $a, a^2, a^3, \cdots a^{p-1}$. They are $p-1$ in number.
I am not satisfied with my above argument. I need more significant proof. Please help me.