Find all homomorphisms from $D_{2n}$ to $\mathbb C^\times$ (revisit)

I actually was asking the same question in here but haven't gotten any feedback yet. I now can elaborate a little so that final answer would be closer. I wanted to find all homomorphisms from the dihedral group $D_{2n}=\langle s,r : s^2=r^n=1 , srs=r^{-1} \rangle$ to the multiplicative group $\mathbb C^\times$. Note that $D_{2n}$ which is generated by $r, s$ so in order to specify a homomorphism from $D_{2n}$ to $\mathbb{C}^\times$, we only need to say what happens to each of these elements. Assume that $\phi:D_{2n}\rightarrow\mathbb{C}^\times$ is a homomorphism, then $\phi(r)^n=1$, so $\phi(r)= \exp(2k\pi i/n)$, $k=0,\ldots,n-1$, and similarly $\phi(s)=\pm 1$. I was aware of that I couldn't conclude there are $2n$ options for homomorphism $\phi$ because for instance $rs$ is also an element of order $2$, then $\phi(rs)=\phi(r)\phi(s)\in \{-1,1\}$. But that means $\phi(r)\in \{-1,1\}$. So if $n$ is odd then $\phi(r) =1$ thus we have $2$ homomorphisms. If $n$ is even then $\phi(r)\in \{-1,1\}$ thus we have $4$ homomorphisms. To ensure that these are all homomorphisms that we need to find, we need straightforward check to show that each of cases indeed gives homomorphism.

Do the above solution looks fine? Any help would much be appreciated.

1 Answer

Nicely done!${}{}{}{}{}{}{}{}$