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

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Statement

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).

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