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 [arXiv:1504.07618, 10.1017/fms.2015.33, MR:3482279].

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- Review status: beta
- Last edited by Andrew Sutherland on 2021-09-18 09:48:48

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