Isomorphism between linear groups when degree power map is bijective

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Statement

Statement for an arbitrary field

Suppose K is a field and n is a natural number such that the map x \mapsto x^n is a bijection from K to itself. Then, the following are true:

  1. The general linear group GL(n,K) is an internal direct product of these two subgroups: the special linear group SL(n,K) and the center Z(GL(n,K)), which is the group of scalar matrices and is isomorphic to the multiplicative group K^*.
  2. The composite of the inclusion of SL(n,K) in GL(n,K) and the quotient map from GL(n,K) to the projective general linear group PGL(n,K) is an isomorphism.
  3. In particular, from (2), we get that SL(n,K) \cong PGL(n,K) \cong PSL(n,K).

Statement for a finite field

Suppose q is a prime power and n is a natural number such that \operatorname{gcd}(n,q - 1) = 1. Then the field \mathbb{F}_q of size q satisfies the condition that the map x \mapsto x^n is a bijection, and the following are true:

  1. The general linear group GL(n,q) is an internal direct product of these two subgroups: the special linear group SL(n,q) and the center Z(GL(n,q)), which is the group of scalar matrices and is isomorphic to the multiplicative group \mathbb{F}_q^*, which is a cyclic group of order q - 1 (see multiplicative group of a finite field is cyclic).
  2. The composite of the inclusion of SL(n,q) in GL(n,q) and the quotient map from GL(n,q) to the projective general linear group PGL(n,q) is an isomorphism.
  3. In particular, from (2), we get that SL(n,q) \cong PGL(n,q) \cong PSL(n,q).

Particular cases

n Congruence condition on q for the isomorphisms to hold (basically, need \operatorname{gcd}(n,q-1) = 1) Initial examples of q
1 all q 2,3,4,5,7,\dots
2 q is even 2,4,8,16,\dots:
SL(2,2) \cong PGL(2,2) \cong PSL(2,2) (all are isomorphic to symmetric group:S3)
SL(2,4) \cong PGL(2,4) \cong PSL(2,4) (all are isomorphic to alternating group:A5)
SL(2,8) \cong PGL(2,8) \cong PSL(2,8) (see projective special linear group:PSL(2,8))
3 q is 0 or -1 mod 3 2,3,5,8,9,11,\dots:
SL(3,2) \cong PGL(3,2) \cong PSL(3,2) (see projective special linear group:PSL(3,2)
SL(3,3) \cong PGL(3,3) \cong PSL(3,3) (see projective special linear group:PSL(3,3))
4 q is even 2,4,8,16,\dots
5 q is not 1 mod 5 2,3,4,5,7,8,9,13,\dots

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