How to solve the exercise: "Find out if the groups $(\mathbb{Z}_2\times\mathbb{Z}_2,+)$ and $(\mathbb{Z}_4,+)$ can or not form an isomorphism"?

Question

I need to find out if the groups $(\mathbb{Z}_2\times\mathbb{Z}_2,+)$ and $(\mathbb{Z}_4,+)$ can or not form an isomorphism. I know that you need to find a bijective function that is also a morphism, but I do not have any idea how to show if there is or not any function. Can you help me?

Answer 1

Any homomorphism from $\mathbb{Z}_4$ is determined by where it sends a generator for the cyclic group. So, where can you send $1\in \mathbb{Z}_4$?

Answer 2

When two groups are isomorphic (let's say there's an isomorphism $f\colon G\cong H$), it means they're essentially the same group; the elements of $G$ and $H$ may have different names, but they'll have the same structural properties. For example, if $a,\,b\in G$ and $ab=ba$, then $f(a)\,f(b)=f(b)\,f(a)$, and vice versa because $f^{-1}$ is also a group homomorphism. If $a\in G$ where $a^n = e$, then $f(a)^n=e$ in $H$, and vice versa; so $a$ and $f(a)$ have the same order (when $f$ is an isomorphism).

Hence the hints in the other two answers: $\mathbb{Z}\mathop{/}4$ has an element $1$ of order 4, so its counterpart in $\left(\mathbb{Z}\mathop{/}2\right)^2$ under any isomorphism $f$ must also have order 4. Is there any such element?