The projective special linear groups $PSL(2,4)$, $PSL(2,5)$ and $PSL(2,9)$ have the property that their orders equal the order of an alternating group. They are also isomorphic to the respective alternating groups.
In this case, we have that $|PSL(2,4)| = |PSL(2,5)| = |A_5|\ $ and $|PSL(2,9)| = |A_6|$. Let $F$ be a finite field. The order of $|PSL(2,F)|$ is given by $(2^n - 1)2^n(2^n + 1)$ when $F$ is of characteristic $2$. Otherwise it is equal to $\frac{1}{2}(p^n - 1)p^n(p^n + 1)$, where $p$ is the characteristic of $F$.
I've been wondering about the following question: when is $|PSL(2,F)| = |A_k|$? In other words, for which $n$ and $k$ the equations
\begin{align*} & 2^{n+1}(2^n - 1)(2^n + 1) = k!\\ &(p^n - 1)p^n(p^n + 1) = k! \text{, where p is an odd prime} \end{align*}
have solutions? Are there only finitely many solutions? And to generalize, what about $PSL(m, F)$?