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

From Groupprops

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 , the projective general linear group of degree two over a finite field of size a prime power . We denote by the underlying prime of , so where .

## Particular cases

## Endomorphism structure

### Automorphism structure

Construct | Value | Order | Comment |
---|---|---|---|

automorphism group | projective semilinear group of degree two | Semidirect product where is cyclic and generated by the Frobenius . | |

inner automorphism group | itself | The group is a centerless group hence equals its inner automorphism group. | |

outer automorphism group | The outer automorphism group is trivial, and hence, the group is a complete group (note that it's already centerless) in the case , i.e., in the case that . 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. |