show · gl2.nonsplit_cartan all knowls · up · search:

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

Authors:
Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2021-09-18 14:51:58
Referred to by:
History: (expand/hide all) Differences (show/hide)