A maximal subgroup of $\GL(2,\F_p)$ that fixes a one-dimensional subspace of $\F_p^2$ is called a **Borel subgroup**. Every Borel 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 [MR:3482279
].

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- Review status: beta
- Last edited by Andrew Sutherland on 2017-03-16 14:01:39

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