Difficult# Order of a Group: Order of Matrix Groups Over Finite Fields

ABSALG-SZGE7B

Let $F_p$, for $p$ prime, be the finite field of $p$ elements.

Let ${\rm GL}_n(p)$ be the group of invertible $n\times n$ matrices with entries in $F_p$.

Then ${\rm GL}_n(p)$ acts on $F_p^n$, the vectors with $n$ components, and with each component in $F_p$.

The group ${\rm SL}_n(p)$ is the subgroup of ${\rm GL}_n(p)$ consisting of those matrices with determinant equal $1$.

The group ${\rm PGL}_n(p)={\rm GL}_n(p)/Z_G$, where $Z_G$ is the center of ${\rm GL}_n(p)$,

and ${\rm PSL}_n(p)={\rm SL}_n(p)/Z_S$, where $Z_S$ is the center of ${\rm SL}_n(p)$.