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For $p>2$ the normalizer of a non-split Cartan subgroup of $\GL_2(\F_p)$ is a maximal subgroup of $\GL_2(\F_p)$ that contains a non-split Cartan subgroup with index 2, and it is the normalizer in $\GL_2(\F_p)$ of the non-split Cartan subgroup it contains. For $p=2$ the normalizer of a non-split Cartan subgroup is defined to be all of $\GL_2(\F_2)$, which contains its (already normal) non-split Cartan subgroup with index 2.

For $p>2$ the label Nn identifies the normalizer of the nonsplit Cartan subgroup generated by the non-split Cartan subgroup Cn and the matrix \[ \begin{pmatrix}1&0\\0&-1\end{pmatrix}, \] and every normalizer of a non-split Cartan subgroup is conjugate to the group Nn.

The label Nn.a.b denotes the proper subgroup of the normalizer of the nonsplit Cartan subgroup Nn generated by the matrices \[ \begin{pmatrix}a&\varepsilon b\\b&a\end{pmatrix}, \begin{pmatrix}1&0\\0&-1\end{pmatrix}. \] where $a$ and $b$ are minimally chosen positive integers and $\varepsilon$ is the least positive integer generating $(\Z/p\Z)^\times\simeq \F_p^\times$, as defined in [MR:3482279] .

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  • Review status: beta
  • Last edited by Andrew Sutherland on 2017-03-16 14:57:18
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