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

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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.
View endomorphism structure of group families | View other specific information about projective general linear group of degree two | View other specific information about group families for rings of the type finite field

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_pq.

Particular cases

Group p q r = \log_pq 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\Gamma L(2,q) r(q^3 - q) Semidirect product PGL(2,q) \rtimes \operatorname{Aut}(\mathbb{F}_q) where \operatorname{Aut}(\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 \operatorname{Aut}(\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.