In Bjorn Poonen's book, Rational Points on Varieties, page 125, in ``Warning 5.1.19", my understanding of what is stated is that over a field of characteristic $p$, the homomorphism from the trivial group scheme to $\mu_p$ (the $p$-th roots of unity of $\mathbb{F}_p$, i.e. the group scheme $\operatorname{Spec}\mathbb{F}_p[t]/(t^p - 1)$) furnishes a counterexample to the claim that
A homomorphism of group schemes that surjective as a map of topological spaces is surjective as a map of group schemes.
If my understanding is right, I fail to see how this counterexample works - the trivial group scheme has just one point, while $\mu_p$ has $p-1$ points, so it seems like no map set theoretic map from the trivial group scheme to $\mu_p$ can be surjective? Also, what is the map that is being talked about here - is it just the map corresponding to the identity?
I would be very grateful if someone can help with the above queries. Please pardon any stupid misunderstanding on my part, I am new to group schemes.
Thanks.
In characteristic $p$, $\mu_p$ has only one point as a scheme. We have $t^p - 1 = (t - 1)^p$ so $\text{Spec } \mathbb{F}_p[t]/(t - 1)^p$ has a single prime ideal which is $(t - 1)$.
There's a unique homomorphism from the trivial group scheme to any other group scheme and he's talking about that one.