Automorphism of group of $\mathbb{Z}_p^{\oplus n}$

Question

Let $\mathbb{Z}_p^{\oplus n}$ be the direct sum of $n$ copies of $\mathbb{Z}_p$. How do I see that the group of group automorphisms Aut$(\mathbb{Z}_p^{\oplus n})$ has the same order with the group GL$(n,\mathbb{Z}_p)$

Answer 1

Hint: an automorphism of $\mathbb{Z}_p\times\cdots\times\mathbb{Z}_p$ is a linear transformation that is invertible (remember that $\mathbb{Z}_p\times\cdots\times\mathbb{Z}_p$ is a vector space over $\mathbb{Z}_p$).