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A split Cartan subgroup of $\GL_2(\F_p)$ is a Cartan subgroup that is diagonalizable over $\F_p$. Every split Cartan subgroup is conjugate to the subgroup of diagonal matrices, which is isomorphic to $\F_p^\times\times\F_p^\times$.

The label Cs identifies the split Cartan subgroup of diagonal matrices.

The label Cs.a.b identifies the proper subgroup of Cs generated by \[ \begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}b&0\\0&r/b\end{pmatrix}, \] 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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  • Last edited by Andrew Sutherland on 2021-09-18 14:52:28
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