Prove the order of each nonindentity element of $G$ is $p$. [closed]
Try thinking about it like this.
Every element of the group $G$ has order dividing $|G| = p^2$.
There are three positive integers dividing $p^2$. What are they?
How many elements have order $1$? Can any elements have order $p^2$?
What can you conclude about the orders of the remaining elements of $G$?