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For $p>2$ the normalizer of a Cartan subgroup of $\GL_2(\F_p)$ is a maximal subgroup of $\GL_2(\F_p)$ that contains a Cartan subgroup with index 2. It is the normalizer in $\GL_2(\F_p)$ of the Cartan subgroup it contains.

For $p=2$ the Cartan subgroups of $\GL_2(\F_2)$ are already normal and we instead define the normalizer of a Cartan subgroup to be a group that contains a Cartan subgroup with index 2. This means that the normalizer of a split Cartan subgroup of $\GL_2(\F_2)$ has order 2 (which makes it conjugate to the Borel subgroup), while the normalizer of a non-split Cartan subgroup of $\GL_2(\F_2)$ has order 6 (which makes it all of $\GL_2(\F_2)$).

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• Review status: beta
• Last edited by Andrew Sutherland on 2017-03-16 15:04:32
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