Prove that if $g$ and $\bar{g}$ commute then so do $\alpha(g)$ and $\alpha(\bar{g})$.
Let $\alpha : G \to H$ is a homomorphism. Let $g, \bar{g} \in G$.
Here's what I have done -
$g\bar{g} = \bar{g}g$
$\alpha(g\bar{g}) = \alpha(\bar{g}g)$
$\alpha(g)\alpha(\bar{g}) = \alpha(\bar{g})\alpha(g)$
Is that it, it seems a bit too trivial?