Existence of nontrivial homomorphism from $Z_3$ to $\text{Aut}(Z_5 \times Z_5)$

Question

Consider $Z_3$, the cyclic group of order $3$.

Now, we know that $|Aut(Z_5 \times Z_5)|=(5^{2}-1)(5^{2}-5)=480$.

So, just from knowing the orders of $Z_3$ and $Aut(Z_5 \times Z_5)$, can we conclude that there exists a nontrivial homomorphism from $Z_3$ to $Aut(Z_5 \times Z_5)$?

Since $3$ divides $480$, it's compatible with Lagrange to have a nontrivial homomorphism, but I'm not sure it's enough to ensure the existence of one.

Answer 1

By Cauchy's Theorem, since $3$ divides the order of $\text{Aut}(Z_5 \times Z_5)$ (and as $3$ is prime), there is an element of $\text{Aut}(Z_5 \times Z_5)$ of order $3$. This element generates a cyclic subgroup isomorphic to $Z_3$ within $\text{Aut}(Z_5 \times Z_5)$. Mapping $Z_3$ to this subgroup is a nontrivial homomorphism.

Answer 2

Since 3 divides 480 and $3$ is prime, Cauchy tells you that there's an element of order 3. Sending the generator of $\mathbb Z_3 $ to that element defines your nontrivial homomorphism ,