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

From Groupprops

## Statement

Let be a prime number. The Projective general linear group of degree two (?) over the prime field , i.e., the group , is a Complete group (?): it is a centerless group and every automorphism of it is inner.

Note that is not complete when is a prime power that is not itself a prime -- there are automorphisms of the group arising from Galois automorphisms of the field extension over its prime subfield.

## Particular cases

Prime number | Group | Order of the group (= ) | 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 | ? |