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Let $\phi \colon G \rightarrow H$ be a surjective morphism of linear algebraic groups. Let $T \subset G$ be a maximal torus: how can I prove that $\phi(T)$ is also a maximal torus?

To show that $\phi(T)$ is a torus is quite easy but I cannot find an argument to prove the maximality.

Thanks for any help.

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  • $\begingroup$ Are you thinking in classical language, or in scheme theoretic language? If in classical, this should follow quite easily from the fourth isomorphism theorem (and, in fact, it should also work for scheme theoretic language). $\endgroup$ Commented Jan 10, 2015 at 0:54
  • $\begingroup$ I'm using the language of varieties, not schemes necessarily. But how do I use the isomorphism theorem to get the result? $\endgroup$
    – N.B.
    Commented Jan 10, 2015 at 17:07

1 Answer 1

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This is a fairly deep fact. It can be found on p. 136 of Humphreys' Linear Algebraic Groups, corrected 3rd printing.

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