I am trying to find a surjective homomorphism from the quaternion group $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ to the cyclic group of order $2$, $H = \{e,x\}$.
I am not sure where to get started on this. I certainly need to map $1$ to $e$ since a homomorphism preserves the identity, and I know that I need to map inverses to inverses. As $H$ only has two elements, almost any map I write down that doesn't send every element of $Q_8$ to the identity is surjective as a map of sets, but that doesn't guarantee that the map respects the group operation. If the groups had the same size, I could write down the multiplication tables, expecting the groups to be "the same but for labels" (as in the case of an isomorphism) and that would inform the map, but since the groups are surely not isomorphic, I don't know a way to derive any insight from the multiplication table.
I would greatly appreciate some hints and some help on how to get started.
$Q_8$ has a subgroup of order four, $\langle i\rangle $. (Actually it has three. ) It's also normal, because of index $2$.
Thus you can simply take the canonical projection: $$\pi:Q_8\to Q_8/\langle i\rangle\cong\Bbb Z_2 $$.