How does one determine the number of group homomorphism from $ D_{8}$ to $\Bbb Q^{\times}$?
For any homomorphism $f$, $o(f(x))$ divides $ o(x) $, in case of both being finite.
Here $\Bbb Q^{\times}$ has only two elements with finite order while each element in $ D_{8} $ is of finite order.
And in $ D_{8} $, for generators $y$ and $x$, we have $yx = xy^{-1}$ where $ y^{2}= e = x^{4}$ .
I need a hint to proceed further.