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

## Contents

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 $q$, the automorphism group of the projective special linear group of degree three $PSL(3,q)$ over the finite field $\mathbb{F}_q$ is the projective outer semilinear group of degree three $PO\Gamma L(3,q)$.

Let $q = p^r$ where $p$ is the underlying prime. The information is presented below:

Construct Value Order Comment
automorphism group projective outer semilinear group of degree three $PO\Gamma L(3,q)$ $2rq^3(q^3 - 1)(q^2 - 1)$ When $r = 1$, this is the same as the projective outer linear group of degree three.
inner automorphism group projective special linear group of degree three $PSL(3,q)$ $q^3(q^3 - 1)(q^2 - 1)/\operatorname{gcd}(3,q-1)$ The group is identified with its inner automorphism group because it is a centerless group.
outer automorphism group
Case $q$ is not 1 mod 3: direct product of cyclic group of order 2 and cyclic group of order $r$
Case $q$ is 1 mod 3: Direct product of cyclic group of order 6 and cyclic group of order $r$
$2r \operatorname{gcd}(3, q - 1)$
Case $q$ is not 1 mod 3: $2r$
Case $q$ is 1 mod 3: $6r$

### Other endomorphisms

Projective special linear group is simple, and in particular $PSL(3,q)$ is always simple, so the only endomorphisms are the automorphisms and the trivial endomorphism.