Isomorphism between linear groups when degree power map is bijective

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	$2,3,4,5,7,8,9,13,\dots$

Related facts

• Isomorphism between linear groups over field:F2