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

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

Can someone help me please? Thank you all

Joe
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  • You know that $C_p\cong \mathbb{Z}/p\mathbb{Z}$. Then given a $\phi$, look here $\phi$ takes $(1,0)$ and $(0,1)$. If you know this can you build a matrix in $GL_2(\mathbb{Z}/p\mathbb{Z})$? Do these matrices classify all the automorphism? – Atticus Christensen May 22 '14 at 16:50
  • All clear, thank you! – Joe May 22 '14 at 17:09
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    The point is that an automorphism of $C_p \times C_p$ as an abelian group is automatically an automorphism of it as an $\mathbb{F}_p$-vector space, or said another way, having an $\mathbb{F}_p$-vector space structure is in fact a property rather than a structure. – Qiaochu Yuan May 23 '14 at 00:54

3 Answers3

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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.

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

WLOG
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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.

Wei Zhou
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