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Let $f:G\to H$ be an isogeny between connected linear algebraic groups. What invariants (rank, semisimple rank, reductive rank, being semisimple...) do these groups share? Are there any properties that one might like these groups to share that they do not? I'm interested in cases when $G$ and $H$ are variously reductive and semisimple so if you need to assume something like this it's no problem.

The motivation is a proof I am reading in which the author seems to be asserting that in such a situation $G$ inherits various properties from $H$.

Mike B
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1 Answers1

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For clarity, an isogeny is a surjective morphism of algebraic groups with finite kernel.

Since an isogeny $f$ induces an isomorphism of Lie algebras, any property of the group that is really a property of the Lie algebra is evidently invariant. So things like being semisimple and being reductive are invariant by isogeny, as are semisimple and reductive ranks. The fundamental example of a (necessary global) characteristic that is not invariant is the fundamental group. Evidently, if $G$ is simply connected the fundamental group of $H$ is $\mathrm{ker}(f)$.

Stephen
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