Let $Q_{8} = \{\pm 1, \pm i, \pm j, \pm k\} \subset \mathbb{H}$ be the quaternion group. How many distinct homomorphisms are there $Q_{8} \rightarrow \{\pm 1 \} \subset \mathbb{R}$?
I can't seem to find one, is there something obvious I'm missing? I'm trying to show that there are four ways of making $\mathbb{R}$ an $\mathbb{R}[Q_{8}]$ module and this seemed to stop me in my path.
If $f:Q_8 \to C_2$ is not trivial then its image is all of $C_2$. Then by the first isomorphism theorem, $f$ factors through an induced isomorphism $Q_8/\ker(f) \cong C_2$ and is essentially the quotient map $Q_8 \to Q_8/\ker(f)$. This maps all of $\ker(f)$ to the identity and the other coset to the nontrivial element. There are three such quotient maps corresponding to the three index two subgroups in $Q_8$. So for instance mapping the whole index two subgroup $\{\pm1,\pm i\}$ to the identity, and the other four elements to the non-identity, yields a nontrivial homomorphism. Similarly for the other two index two subgroups.