Endomorphism structure of projective general linear group of degree two over a finite field

This article gives specific information, namely, endomorphism structure, about a family of groups, namely: projective general linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.
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This page describes the endomorphism structure of $PGL(2,q)$, the projective general linear group of degree two over a finite field of size a prime power $q$. We denote by $p$ the underlying prime of $q$, so $q=p^r$ where $r={\log }_{p}q$.

Particular cases

Group	$p$	$q$	$r={\log }_{p}q$	Order of the group $=q^4-q^3-q^2+q$	Automorphism group	Order of the automorphism group $=r(q^3-q)$	Order of inner automorphism group ($=q^3-q$)	Order of outer automorphism group, same as the Galois group ($=r$)	Endomorphism structure page
symmetric group:S3	2	2	1	6	symmetric group:S3	6	6	1	endomorphism structure of symmetric group:S3
symmetric group:S4	3	3	1	48	symmetric group:S4	24	24	1	endomorphism structure of symmetric group:S4
alternating group:A5	2	4	2	60	symmetric group:S5	120	60	2	endomorphism structure of alternating group:A5
symmetric group:S5	5	5	1	120	symmetric group:S5	120	120	1	endomorphism structure of symmetric group:S5
projective general linear group:PGL(2,7)	7	7	1	336	projective general linear group:PGL(2,7)	336	336	1	endomorphism structure of projective general linear group:PGL(2,7)
projective special linear group:PSL(2,8)	2	8	3	504	Ree group:Ree(3)	1512	504	3	endomorphism structure of projective special linear group:PSL(2,8)
projective general linear group:PGL(2,9)	3	9	2	720	automorphism group of alternating group:A6	1440	720	2	endomorphism structure of projective general linear group:PGL(2,9)

Endomorphism structure

Automorphism structure

Construct	Value	Order	Comment
automorphism group	projective semilinear group of degree two $P\mathrm{\Gamma }L(2,q)$	$r(q^3-q)$	Semidirect product $PGL(2,q)⋊\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{F}}_q)$ where $\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{F}}_q)\cong \mathbb{Z}/r\mathbb{Z}$ is cyclic and generated by the Frobenius $x\mapsto x^p$.
inner automorphism group	$PGL(2,q)$ itself	$q^3-q$	The group is a centerless group hence equals its inner automorphism group.
outer automorphism group	$\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{F}}_q)\cong \mathbb{Z}/r\mathbb{Z}$	$r$	The outer automorphism group is trivial, and hence, the group is a complete group (note that it's already centerless) in the case $r=1$, i.e., in the case that $q=p$. In other words, a projective general linear group of degree two over a finite field is complete if and only if that field is a prime field.