Automorphism of Direct Product $Z_3\times Z_3$

Question

I know form Show $\mid Aut(Z_3 \times Z_3) \mid=48$ that order of Automorphism of $Z_3\times Z_3$ is 48 .And which is same as order of $GL_2(F_3)$.But I wanted to show they are isomorphic .How should I approach this problem?
Any Help Will be appreciated.

Answer 1

Think of $Z_3$ additively, then your group is $\mathbb{F}_3\oplus\mathbb{F}_3$.

Now check that there is one and only one way to make this additive group into an $\mathbb{F}_3$ vector space, we have to define $0\cdot x=0$, we have to define $1\cdot x=x$ and so we have to define $2\cdot x=(1+1)x=x+x$. Check this is indeed a vector space.

Now check that any automorphism of the additive group must also be an automorphism of the vector space. For instance, $$2\cdot \alpha(x)=\alpha(x)+\alpha(x)=\alpha(x+x)=\alpha(2\cdot x).$$

There is no difficulty showing that an automorphism of the vector space is an automorphism of the additive group.

Hence the automorphism group of $Z_3\times Z_3$ is the autmorphism group of $\mathbb{V}_2(\mathbb{F}_3)$, that is $\text{GL}(2,\mathbb{F}_3)$.