I am looking for a subgroup of order $n$ of $GL_n(\mathbb{F}_q)$ where $q=p^a$ with $p$ prime and $a\ge 1$.
I know that $SL_2(\mathbb{F}_2)$ is isomorphic to $\mathcal{A}_3$ but it does not fit with the statement...
Thanks in advance !
2026-03-25 23:39:23.1774481963
Subgroup of order $n$ of $GL_n(\mathbb{F}_q)$
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The subgroup generated by the cyclic permutation matrix $$\left(\begin{matrix}0 & 0 & 0& \cdots & 0& 0 & 1\\ 1 & 0& 0 & \cdots & 0 & 0& 0\\ 0 & 1& 0 & \cdots & 0 & 0& 0\\ 0 & 0 &1& \cdots & 0 & 0& 0\\ \vdots &\vdots &\vdots & &\vdots &\vdots &\vdots \\ 0& 0 & 0 & \cdots & 1 & 0& 0\\ 0 & 0 & 0& \cdots & 0&1 & 0\\ \end{matrix}\right)$$
has the same order as the size of the matrix. This does not depend on the field where the coefficients live.