What is the cardinality of the group generated by $$\begin{pmatrix}0&-1\\1&0\end{pmatrix},\begin{pmatrix}0&1\\-1&-1\end{pmatrix}$$ under multiplication?
2026-03-25 05:06:44.1774415204
Cardinality of group generated by two matrices
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The subgroup generated by two elements $g$ and $h$ is the set of all words in $\{g,h,g^{-1},h^{-1} \}$, which is countable.
Now $$ \begin{pmatrix}0&-1\\1&0\end{pmatrix} \begin{pmatrix}0&1\\-1&-1\end{pmatrix} = \begin{pmatrix}1&1\\0&1\end{pmatrix} $$ and $$ \begin{pmatrix}1&1\\0&1\end{pmatrix}^n= \begin{pmatrix}1&n\\0&1\end{pmatrix} $$ Thus, the subgroup is infinite.