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!
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.