# Projective general linear group of degree two over a prime field is complete

## Statement

Let $p$ be a prime number. The Projective general linear group of degree two (?) over the prime field $\mathbb{F}_p$, i.e., the group $PGL(2,p)$, is a Complete group (?): it is a centerless group and every automorphism of it is inner.

Note that $PGL(2,q)$ is not complete when $q$ is a prime power that is not itself a prime -- there are automorphisms of the group arising from Galois automorphisms of the field extension $\mathbb{F}_q$ over its prime subfield.

## Particular cases

Prime number $p$ Group $PGL(2,p)$ Order of the group (= $p^3 - p$) Other ways of seeing that the group is complete
2 symmetric group:S3 6 symmetric groups on finite sets are complete (with the exception of degrees 2 and 6, here the degree is 3)
3 symmetric group:S4 24 symmetric groups on finite sets are complete (with the exception of degrees 2 and 6, here the degree is 4)
5 symmetric group:S5 120 symmetric groups on finite sets are complete (with the exception of degrees 2 and 6, here the degree is 4)
7 projective general linear group:PGL(2,7) 336  ?