Projective special linear group equals projective general linear group iff nth power map is surjective

Statement
Suppose $$G = GL(n,k)$$ is the fact about::general linear group over a field $$k$$, $$Z$$ is the subgroup of $$G$$ comprising the scalar matrices, and $$\rho:G \to G/Z$$ is the quotient map. $$G/Z = PGL(n,k)$$ is the fact about::projective general linear group, and if $$S \le G$$ is the fact about::special linear group, $$\rho(S)$$ is the fact about::projective special linear group.

Then, $$\rho(S) = \rho(G)$$ if and only if the map $$x \mapsto x^n$$ is a surjective map from $$k$$ to itself. In particular:


 * For a prime power $$q$$, $$PSL(n,q) = PGL(n,q)$$ if and only if $$n$$ is relatively prime to $$q - 1$$.
 * For $$k$$ an algebraically closed field, $$PSL(n,k) = PGL(n,k)$$.

Related facts

 * Special linear group is cocentral in general linear group iff nth power map is surjective
 * Center of general linear group is group of scalar matrices over center