My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record
I have two question about Proposition 3.3.:
Proposition3.3. says that $1 \rightarrow Aut(E) \xrightarrow{j} G(E) \xrightarrow{p} T \rightarrow 1 $ is a short exact sequence of algebraic groups over an algebraically closed field $k$, where T is a torus over $k$ and $E$ is a vector bundle of a almost homogeneous variety $X$. I have no question about exactness, but the author says that since both $Aut(E) $ and $T$ are linear algebraic groups, $G(E)$ is a linear algebraic group.
Question1.
Is that means that if $1 \rightarrow G \xrightarrow{f} H \xrightarrow{g} K \rightarrow 1 $ is a short exact sequence of algebraic groups, then $H$ is linear if, and only if $G$ and $K$ are linear? Is that easy to see?
In the proof of Proposition3.3., the author says that a surjective algebraic group homomorphism from a torus to a torus always has a section (see Borel [1]).
Question2.
I already found this reference :A. Borel, "Linear algebraic groups", Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201. But I can't find where prove this fact. Who can tell me where is the proof?
Thank you very much!