Are there 2p elements of order p in $\mathbb{Z}_p \times\mathbb{Z}_p$? I know the order of  $\mathbb{Z}_p \times\mathbb{Z}_p$ is $p^2$, that there are $p^2$ elements in the set represented by  $\mathbb{Z}_p \times\mathbb{Z}_p$. Of those elements how many are of order $p$? I'm thinking it is $2p$, since the elements of  $\mathbb{Z}_p \times\mathbb{Z}_p$ are of the form $(a,b)$, where $a$ and $b$ are the elements of order $p$ are of the form $(a,0)$ and $(0,b)$. 

Hint: What is the order of $(1,1)$?
(Hint 2: The only possible orders of an element divide $p^2$ and are hence $1$, $p$ or $p^2$. Can any non-identity element have order $1$ or $p^2$?)
