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A non-split Cartan subgroup of $\GL(2,\F_p)$ is a Cartan subgroup that is not diagonalizable over $\F_p$. Every non-split Cartan subgroup is a cyclic group isomorphic to $\F_{p^2}^\times$.

For $p=2$ the label Cn identifies the unique index 2 subgroup of $\GL(2,\F_2)$. For $p>2$ the label Cn identifies the nonsplit Cartan subgroup consisting of matrices of the form $\begin{pmatrix}x&\varepsilon y\\y&x\end{pmatrix},$ with $x,y\in \F_p$ not both zero and $\varepsilon$ the least positive integer generating $(\Z/p\Z)^\times\simeq \F_p^\times$, corresponding to $x+y\sqrt{\varepsilon}\in\F_{p^2}^\times$. Every non-split Cartan subgroup is conjugate to the group Cn.

Labels of the form Cn.a.b identify the proper subgroup of Cn generated by the matrix $\begin{pmatrix}a&\varepsilon b\\b&a\end{pmatrix},$ 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 14:27:45
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