How do we find the homorphism from $\mathbb{Z_2} \to{\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$

Question

How do we find the homorphism from $\mathbb{Z_2} \to {\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})?$

I know that ${\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ is isomorphic to $GL_2(\mathbb{Z_3})$.

We know that $\theta :\mathbb{Z_2} \to GL_2(\mathbb{Z_3}) $

Then $\theta(1)$ is mapped to an element of order 2 which should be present in the Sylow - $2$ subgroup of $GL_2(Z_3)$.

Then what automorphism map corresponds to the Sylow $2$ subgroup? What is the structure of the Sylow $2$ subgroup?

Answer 1

By Cauchy's Theorem, since $2\mid |GL_2(\Bbb Z_3)|=48$, there exists at least one element $g\in GL_2(\Bbb Z_3)$ of order two. Now

$$\begin{align} \varphi:\Bbb Z_2&\to GL_2(\Bbb Z_3),\\ z&\mapsto g,\\ e&\mapsto \begin{pmatrix} 1&0\\ 0&1\end{pmatrix} \end{align}$$

is a homomorphism, where $z$ is the nontrivial element of $\Bbb Z_2$.