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

Suppose $G = GL(n,k)$ is the 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 Projective general linear group (?), and if $S \le G$ is the Special linear group (?), $\rho(S)$ is the 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, $PSL(n,k) = PGL(n,k)$.