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I remember recently seeing, but I have no memory of where, a result of the following form:

Let $k$ be a field, $G$ a connected reductive $k$-group, and $H$ a subgroup of $G$ such that …. Then $H$ normalizes a maximal torus in $G$.

The hypothesis on $H$ is suppressed, and here's why. At the time, I thought "I'm sure that result will come in handy, so I'd better remember it." Now, of course, I need such a result, and so I no longer remember it. Presumably there are many possible things that can fill in the blank, including the annoyingly tautological ("such that $H$ normalizes a maximal torus in $G$"). What are the most interesting ways to fill it in that you know?

There's no harm in assuming $k$ is algebraically closed, but I would prefer not to assume that it has characteristic $0$. I'm most interested in the case where $H$ is finite, and even étale, but you need not assume that.

LSpice
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1 Answers1

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Browsing Lie-group questions turned up a comment from @abx on MO that may have been what I remembered. I have asked them to post their comments as an answer, but, in the meantime, I record this as a reminder, in case I forget again.

Another approach: any abelian (or even nilpotent) subgroup of $G$ is contained in the normalizer of a maximal torus.

Here $G$ is a compact Lie group, but perhaps there is an analogous result for a reductive group over an algebraically closed field of characteristic $0$. (I originally said "maybe the same is true", but a colleague pointed out to me that it can't be literally the same result; the centre of the unipotent radical of a Borel subgroup of a non-toral reductive group is Abelian, but will not normalise a maximal torus.) @abx also gave a reference:

Bourbaki's Lie IX, §5, Cor. 4 of Théorème 1.

For my reference, this is in no. 4 of §5, on the page labelled LIE IX.49 in my copy.

LSpice
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