Why is $\bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$ not isomorphic to $\mathbb{Z}_2 \times \bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$

Question

I was wondering if someone could offer a hint on how to show that $\bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$ is not isomorphic to $\mathbb{Z}_2 \times \bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$?

Here are a few observations I made

1. They both have the same cardinality.
2. Both of these groups are not cyclic.
3. The nonidentity elements in $\bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$ have either order $2$ or $4$. It appears that this is also the case for $\mathbb{Z}_2 \times \bigoplus_{i \in \mathbb{N}} \mathbb{Z}_4$.
4. I attempted to prove by contradiction by assuming there was an isomorphism $\varphi$, and tried to find a formula for it, but got nowhere with that; and I am unsure on how to proceed with contradiction.

Am I missing something obvious here? I'd appreciate a hint

Thank you for taking the time to read this!

Edited Changed $Z_4$ to $\mathbb{Z}_4$

Answer 1

Let $\Phi$ be the (first-order logic) assertion $$\forall x,\;x^2=1 \Rightarrow\exists y:\; y^2=x$$

Then $(\mathbf{Z}/4\mathbf{Z})^{(I)}$ satisfies $\Phi$ but $G\times\mathbf{Z}/2\mathbf{Z}$ doesn't for any group $G$.

Hence your two groups are not isomorphic (they are not even elementary equivalent).