Separating invariants over finite fields

We determine the minimal number of separating invariants for the invariant ring of a matrix group $G<{\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_q)$ over the finite field ${\mathbb{F}}_q$. We show that this minimal number can be obtained with invariants of degree at most $\mathrm{\mid }G\mathrm{\mid }n(q-1)$. In the non-modular case this construction can be improved to give invariants of degree at most $n(q-1)$. As examples we study separating invariants over the field ${\mathbb{F}}_2$ for two important representations of the symmetric group