Is exponentiation of a fixed generator modulo a prime collision-resistant?

Question

Let $p$ be a prime number, and $g$ a generator of $\mathbb Z/p\mathbb Z$. For a message $m$, define the hash function $$h(m) = g^m \pmod p.$$ Is $h$ collision-resistant?

Answer 1

Let $m$ be arbitrary. Then $m'=m+p-1$ yields a collision with $h(m)=h(m')$ as $m\equiv m'\pmod{p-1}$ and thus by $p-1$ being the relevant group's order $g^m\equiv g^{m'}\pmod p$.

Or formulated differently (using $g^{p-1}\bmod p=1$): $$h(m')=g^{m+p-1}=g^m\underbrace{g^{p-1}}_{1}\equiv g^m=h(m)\pmod p$$