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A Borel subgroup of a general linear group is a subgroup that is conjugate to the group of upper triangular matrices.

The Borel subgroups of $\GL_2(\F_p)$ are maximal subgroups that fix a one-dimensional subspace of $\F_p^2$; every such subgroup is conjugate to the subgroup of upper triangular matrices.

Subgroup labels containing the letter B identify a subgroup of $\GL_2(\F_p)$ that lies in the Borel subgroup of upper triangular matrices but is not contained in the subgroup of diagonal matrices; these are precisely the subgroups of a Borel subgroup that contain an element of order $p$.

The label B is used for the full Borel subgroup of upper triangular matrices

The label B.a.b denotes the proper subgroup of B generated by the matrices \[ \begin{pmatrix}a&0\\0&1/a\end{pmatrix},\ \begin{pmatrix}b&0\\0&r/b\end{pmatrix},\ \begin{pmatrix}1&1\\0&1\end{pmatrix}, \]
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].

Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2021-09-18 13:50:44
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