# Endomorphism structure of projective special 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 special linear group of degree two.

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

## Endomorphism structure

### Automorphism structure

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

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

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

automorphism group | projective semilinear group of degree two | When , i.e., the field is a prime field, then the automorphism group is just . | |

inner automorphism group | projective special linear group of degree two | The group is identified with its inner automorphism group because it is a centerless group. The order is if is even, and if is odd. | |

outer automorphism group | Case even: cyclic group of order Case odd: Direct product of cyclic group of order 2 and cyclic group of order |
Case even: Case odd: |

### Other endomorphisms

If is 4 or more, the group PSL(2,q) is simple, so the only endomorphisms are the trivial endomorphism and the automorphisms. If (giving symmetric group:S3) or (giving alternating group:A4) then there are other endomorphisms with nontrivial kernels.