Does every non-trivial element of $\mathbb{Z}_p$ generate the group? This just popped up in my head and I just wanted to make sure if I'm right.
Every element (except the identity element $0$) of the group $\mathbb{Z}_p$ (under addition and $p$ is prime) is a generator for the group. For example, $\mathbb{Z}_5 = \langle 1 \rangle = \langle 2 \rangle = \langle 3 \rangle = \langle 4 \rangle$.
Thanks!

You can show that $\mathbb{Z}_p$ is a field, so if $H= \langle h \rangle$ with $h \neq 0$, $1 \in H$ since $h$ is invertible; you deduce that $H= \mathbb{Z}_p$, ie. $h$ is a generator.
