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

## Statement

Suppose is the General linear group (?) over a field , is the subgroup of comprising the scalar matrices, and is the quotient map. is the Projective general linear group (?), and if is the Special linear group (?), is the Projective special linear group (?).

Then, if and only if the map is a surjective map from to itself. In particular:

- For a prime power , if and only if is relatively prime to .
- For an algebraically closed field, .