How to Find $\operatorname{Aut}(\mathbb Z_8 × \mathbb Z_8)$?

Question

I need to find a group that is isomorphic to $\operatorname{Aut}(\mathbb Z_8 × \mathbb Z_8)$.

In case there's a product of more than two groups, how could I find its automorphic object? Is there any typical algorithm to find one?

Answer 1

I presume $Z8=\Bbb Z/8\Bbb Z$. In this case the automorphism group of $\Bbb Z/8\Bbb Z\times\Bbb Z/8\Bbb Z$ is $\text{GL}_2(\Bbb Z/8\Bbb Z)$ the group of two by two invertible matrices over $\Bbb Z/8\Bbb Z$. In general the automorphism group of a direct product of $n$ copies of $\Bbb Z/m\Bbb Z$ is $\text{GL}_n(\Bbb Z/m\Bbb Z)$.

In general, there won't always be a simple description of the automorphism group of a direct product of groups.