arithmetic geometry - Torsion of an abelian variety under reduction.

This is Hartshorne, Exercise IV.4.19: Let $mathcal{A}/mathbb{Z} setminus S =: T$ be an Abelian scheme. The multiplication by $n$ morphism is flat [I don't know how to show this, but I think it can be found in Katz-Mazur.], so the $n$-Torsion $mathcal{A}[n] to mathcal{A}$ is also flat as it is a base change and $mathcal{A} to T$ also since it is flat. It is also proper and quasi-finite, and therefore finite. So we have a finite flat group scheme. Because of $(n,p) = 1$ $mathcal{A}[n]$ is étale over $mathbb{Z}_{(p)}$ (how to show this?). We have for $X/S$ finite étale

Consider the reduction map $mathcal{A}[n]_eta(mathbf{Q}) = mathcal{A}[n](T) to mathcal{A}[n]_p(mathbf{F}_p)$ for $(n,p) = 1$, confer Liu, Chapter 10.1.3. Liu, Proposition 10.1.40(b) gives us one-point fibres of the reduction map.

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