How can I prove that $Aut(C_p\times C_p)\simeq GL_2(\mathbb Z/p\mathbb Z)$?

Question

How can I prove that $Aut(C_p\times C_p)\simeq GL_2(\mathbb Z/p\mathbb Z)$?

No theorical argument came to my mind, so I'm trying to build explicitly an isomorphism $\phi:Aut(C_p\times C_p)\longrightarrow GL_2(\mathbb Z/p\mathbb Z)$, but I'm stuck.

Can someone help me please? Thank you all

Answer 1

If you write $G=C_p \times C_p$ in addition, you will find each element of $G$ is a linear combination of $a,b$ with coefficient in $F_p$, where $a, b$ are the generators of $G$. Then you can check that a hommormorphism of $G$ (in multiplication) will become linear transformation (in addition). Then you can get your proof.

Answer 2

Hint: $C_p \cong \mathbb{Z}/ p \mathbb{Z}$ is vector space

Answer 3

Hint. Define $\pi_1:(x,y)\mapsto (x,0)$ and $\pi_2:(x,y)\mapsto (0,y)$. Show that an automorphism $\phi:C_p\times C_p\rightarrow C_p\times C_p$ is completely determined by $\pi_1\phi\pi_1$, $\pi_2\phi\pi_1$, $\pi_1\phi\pi_2$, and $\pi_2\phi\pi_2$. Then prove that there is an isomorphism between $\operatorname{Aut}(C_p\times C_p)$ and $\operatorname{GL}_2(\mathbb{Z}/p\mathbb{Z})$ using this information.