I am looking for reference or hints how to prove the following result. 



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> Let $G$ be a commutative $S$-group scheme which is the extension of an abelian scheme $A$ by a torus $T$. Then the n-torsion $G[n]$ is a finite flat $S$-group scheme. 
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Specifically, I have difficulties in showing that $G[n]$ is finite. For a general semi-abelian scheme we know that it is quasi-finite and flat, but not necessarily finite (see e.g. the book Neron Models, Lemma 7.3/2).


Thanks in advance,



Is the n-torsion of an extension of an abelian variety by a torus, finite and flat?
It is an exercise with descent theory and the snake lemma for fppf abelian group sheaves to deduce the result for $G[n]$ from the cases of $T[n]$ and $A[n]$.


In more detail, by the snake lemma $G[n]$ is an extension of $A[n]$ by $T[n]$ in the sense of such abelian sheaves. Since $A[n]$ and $T[n]$ are each finite fppf over $S$, the same then holds for $G[n]$. Indeed, rather generally, if
$$1 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 1$$
is a complex of $S$-group schemes with $G'$ affine fppf over $S$ and the diagram is short exact for the fppf topology (so $G'$ is the scheme-theoretic kernel of $G \rightarrow G''$) then the functor of points of $G$ as a $G''$-scheme is a $G'$-torsor for the fppf topology on $G''$, so the $G''$-scheme $G \rightarrow G''$ becomes isomorphic *fppf-locally on $G''$* to $G'$ (over the base) as a scheme. Hence, by fppf descent for properties of morphisms, $G \rightarrow G''$ inherits many "nice" properties that may be satisfied by $G' \rightarrow S$, such as: proper, flat, smooth, etale, finite, etc. In particular, $G$ is fppf over $G''$ and if $G'$ is finite over $S$ then so is $G \rightarrow G''$ (and hence so is $G$ if $G''$ is 
also finite over $S$).


See Oort's LNM book on commutative group schemes for generalizations with the fpqc topology (around section 18, IIRC).


