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

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

## Endomorphism structure

### Automorphism structure

For any prime power , the automorphism group of the 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 | This is the quotient by the center. The center has order 1 and has order if is even. The center has order 2 and has order 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, SL(2,q) is quasisimple. Further, we have that finite quasisimple implies every endomorphism is trivial or an automorphism. Combining, we get that for , the endomorphisms of are the automorphisms and the trivial endomorphism.

The cases (giving symmetric group:S3 -- see endomorphism structure of symmetric group:S3) or (giving special linear group:SL(2,3) -- see endomorphism structure of special linear group:SL(2,3)) are somewhat different.