# Endomorphism structure of projective special linear group of degree three over a finite field

From Groupprops

This article gives specific information, namely, endomorphism structure, about a family of groups, namely: projective special linear group of degree three.

View endomorphism structure of group families | View other specific information about projective special linear group of degree three

### Automorphism structure

For any prime power , the automorphism group of the projective special linear group of degree three over the finite field is the projective outer semilinear group of degree three .

Let where is the underlying prime. The information is presented below:

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

automorphism group | projective outer semilinear group of degree three | When , this is the same as the projective outer linear group of degree three. | |

inner automorphism group | projective special linear group of degree three | The group is identified with its inner automorphism group because it is a centerless group. | |

outer automorphism group | Case is not 1 mod 3: direct product of cyclic group of order 2 and cyclic group of order Case is 1 mod 3: Direct product of cyclic group of order 6 and cyclic group of order |
Case is not 1 mod 3: Case is 1 mod 3: |

### Other endomorphisms

Projective special linear group is simple, and in particular is always simple, so the only endomorphisms are the automorphisms and the trivial endomorphism.