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

From Groupprops
Jump to: navigation, search

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  ?