Determine all homomorphisms from $D_{5}$ onto $Z_{2} \bigoplus Z_{2}$. Determine all homomorphisms from $D_{5}$ to $Z_{2} \bigoplus Z_{2}$. Determine all homomorphisms from $D_{5}$ onto $Z_{2}\bigoplus Z_{2}$. Determine all homomorphisms from $D_{5}$ to $Z_{2}\bigoplus Z_{2}$.
The first thing, I think, would be to make sure of is that the generators for $D_5$ map to compatible elements. Not sure where to go from there.
Any help would be appreciated!
 A: The first thing you should do is find out what might be the kernel of such a homomorphism. The normal subgroups of $D_5$ are $\{e\},\langle r\rangle$ and $D_5$ when $r$ is the rotation by angle $\frac{2\pi}{5}$ clockwise. I'll leave you to check that these are all the normal subgroups of $D_5$, hence the kernel cannot be anything else. So now split into cases. Can the kernel of a homomorphism be all $D_5$? Yes, if this is the trivial homomorphism. Can the kernel be $\{e\}$? No, because that means the homomorphism is injective but there cannot be an injective function from a set of $10$ elements to a set of $4$ elements. 
So now we only need to find the homomorphisms $\varphi:D_5\to\mathbb{Z_2}\times\mathbb{Z_2}$ where $Ker(\varphi)=\langle r\rangle$. By an isomorphism theorem it follows that $|Im(\varphi)|=2$. All rotations are mapped to the identity and all reflections must be mapped to the same non trivial element of $\mathbb{Z_2}\times\mathbb{Z_2}$. Let $x\in\mathbb{Z_2}\times\mathbb{Z_2}$ be a non trivial element. I'll leave you to check that the function $\varphi:D_5\to\mathbb{Z_2}\times\mathbb{Z_2}$ which maps rotations to the identity and reflections to $x$ is indeed a homomorphism. There are $3$ non trivial elements in $\mathbb{Z_2}\times\mathbb{Z_2}$ so there are $3$ homomorphisms with kernel $\langle r\rangle$. Together with the trivial homomorphism we get there are $4$ homomorphisms in general. 
