The **normalizer of a split Cartan subgroup** of $\GL_2(\F_p)$ is a maximal subgroup of $\GL_2(\F_p)$ that contains a split Cartan subgroup with index 2. For $p>2$ such a group is in fact the normalizer in $\GL_2(\F_p)$ of the split Cartan subgroup it contains, but for $p=2$ this is not the case (the split Cartan subgroup of $\GL_2(\F_2)$ is already normal).

The label **Ns** identifies the subgroup generated by the split Cartan subgroup **Cs** of diagonal matrices and the matrix
\[
\begin{pmatrix}0&1\\1&0\end{pmatrix}.
\]
Every normalizer of a split Cartan subgroup is conjugate to the group **Ns**.

The label **Ns.a.b** identifies the proper subgroup of **Ns** generated by the matrices
\[
\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-r/b&0\end{pmatrix},
\]
where $a$ and $b$ are minimally chosen positive integers and $r$ is the least positive integer generating $(\Z/p\Z)^\times\simeq \F_p^\times$.

The label **Ns.a.b.c** identifies the proper subgroup of the normalizer of the split Cartan subgroup generated by the matrices
\[
\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-1/b&0\end{pmatrix}, \begin{pmatrix}0&c\\-r/c&0\end{pmatrix}
\]
where $a$ and $b$ are minimally chosen positive integers and $r$ 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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**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2021-09-18 09:49:03

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