Let $C_p$ be a cyclic group of prime order $p$. Find the number of the automorphism group of $C_p^{\oplus n}$.
Is this approach correct?
$C_p$ is isomorphic to $(\mathbb Z_p,+)$. The automorphism groups of these two are therefore isomorphic. Further, $\mathbb Z_p$ is also a field. So $\mathbb Z_p^n$ is a vector space of dimension $n$ over the field with $p$ elements. So the number of automorphisms of the group $\mathbb Z_p^n$ is the number of automorphisms of the vector space, which is the number of elements in $GL_n(\mathbb F_p)$. For further calculations, see this question.
