**Proof.**
Part (1) follows from Lemma 39.9.2. Part (2) follows from Lemma 39.9.5. Part (3) follows from Lemma 39.9.8. If $k$ is algebraically closed then surjective morphisms of varieties over $k$ induce surjective maps on $k$-rational points, hence (4) follows from (3). Part (5) follows from Lemma 39.9.8 and the fact that a base change of a finite locally free morphism of degree $N$ is a finite locally free morphism of degree $N$. Part (6) follows from Lemma 39.9.9. Namely, if $n$ is invertible in $k$, then $[n]$ is étale and hence $A[n]$ is étale over $k$. On the other hand, if $n$ is not invertible in $k$, then $[n]$ is not étale at $e$ and it follows that $A[n]$ is not étale over $k$ at $e$ (use Morphisms, Lemmas 29.36.16 and 29.35.15).

Assume $k$ is algebraically closed. Set $g = \dim (A)$. Proof of (7). Let $\ell $ be a prime number which is invertible in $k$. Then we see that

\[ A[\ell ](k) = A(k)[\ell ] \]

is a finite abelian group, annihilated by $\ell $, of order $\ell ^{2g}$. It follows that it is isomorphic to $(\mathbf{Z}/\ell \mathbf{Z})^{2g}$ by the structure theory for finite abelian groups. Next, we consider the short exact sequence

\[ 0 \to A(k)[\ell ] \to A(k)[\ell ^2] \xrightarrow {\ell } A(k)[\ell ] \to 0 \]

Arguing similarly as above we conclude that $A(k)[\ell ^2] \cong (\mathbf{Z}/\ell ^2\mathbf{Z})^{2g}$. By induction on the exponent we find that $A(k)[\ell ^ m] \cong (\mathbf{Z}/\ell ^ m\mathbf{Z})^{2g}$. For composite integers $n$ prime to the characteristic of $k$ we take primary parts and we find the correct shape of the $n$-torsion in $A(k)$. The proof of (8) proceeds in exactly the same way, using that Lemma 39.9.10 gives $A(k)[p] \cong (\mathbf{Z}/p\mathbf{Z})^{\oplus f}$ for some $0 \leq f \leq g$.
$\square$

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