Abstract Algebra: Theory and Applications · Problem 4

Permutation Groups — Thomas Judson

Mathematics — Abstract Algebra · cyclic-groups-matrix-subgroups

Problem

Find the subgroups of

GL_2(\mathbb{R})

generated by each of the following matrices. ((a))

\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

((b))

\begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}

((c))

\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}

((d))

\begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}

((e))

\begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}

((f))

\begin{pmatrix} \sqrt{3}/2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}

Hint. (a)

\begin{pmatrix}1&0\\0&1\end{pmatrix}, \begin{pmatrix}-1&0\\0&-1\end{pmatrix}, \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \begin{pmatrix}0&1\\-1&0\end{pmatrix}

. (c)

\begin{pmatrix}1&0\\0&1\end{pmatrix}, \begin{pmatrix}1&-1\\1&0\end{pmatrix}, \begin{pmatrix}-1&1\\-1&0\end{pmatrix}, \begin{pmatrix}0&1\\-1&1\end{pmatrix}, \begin{pmatrix}0&-1\\1&-1\end{pmatrix}, \begin{pmatrix}-1&0\\0&-1\end{pmatrix}