I have a task where I need to show that there are exactly 4 homomorphisms from $D_4 $ to $ Q_8 $.
I've been trying to use the fact that if $ \varphi: G \rightarrow H $ is a homomorphism, then $ |\varphi(g)|$ is dividing $|g|$.
However, It feels somewhat tedious to check what $\varphi$ can be and to check every single possibility. Is this how I'm supposed to do this?
Keep in mind that I still don't fully understand how to work with group theory, so I'm not really sure if I'm doing the right stuff at all.
Thanks for the help!
Best Answer
Let $Q_8=\{\pm i, \pm j, \pm k , \pm 1\}$, $D_4$ is generated by a rotation $\rho$ of order $4$ and a reflection $r$ of order $2$, the defining relation $r\rho r=\rho^3$. So you want to see where $\rho, r$ can go under a homomorphism $\phi$.
The image of $r$ should have order $2$ or $1$. So it is either $-1$ or $1$. If the image of $r$ is $1$ then $\phi(\rho)=\phi(\rho^3)$, so $\phi(\rho)^2=1$, thus $\phi(\rho)$ is either $-1$ or $1$, we have two such homomorphisms.
Suppose that $\phi(r)=-1$. Then again $\phi(\rho^3)=\phi(r)\phi(\rho)\phi(r)=\phi(\rho)$ and, again, $\phi(\rho)=\pm 1$ and we have two such homomorphisms, total of $4$.