# Endomorphism structure of projective special linear group of degree two 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 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 $q$, the automorphism group of the projective special linear group of degree two $PSL(2,q)$ over the finite field $\mathbb{F}_q$ is the projective semilinear group of degree two $P\Gamma L(2,q)$.

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

Construct Value Order Comment
automorphism group projective semilinear group of degree two $P\Gamma L(2,q)$ $r(q^3 - q)$ When $q = p$, i.e., the field is a prime field, then the automorphism group is just $PGL(2,q)$.
inner automorphism group projective special linear group of degree two $PSL(2,q)$ $(q^3 - q)/\operatorname{gcd}(2,q-1)$ The group is identified with its inner automorphism group because it is a centerless group. The order is $q^3 - q$ if $q$ is even, and $(q^3 - q)/2$ if $q$ is odd.
outer automorphism group Case $q$ even: cyclic group of order $r$
Case $q$ odd: Direct product of cyclic group of order 2 and cyclic group of order $r$
Case $q$ even: $r$
Case $q$ odd: $2r$

### Other endomorphisms

If $q$ is 4 or more, the group PSL(2,q) is simple, so the only endomorphisms are the trivial endomorphism and the automorphisms. If $q = 2$ (giving symmetric group:S3) or $q = 3$ (giving alternating group:A4) then there are other endomorphisms with nontrivial kernels.

## Particular cases

Field size $q$ Field characteristic $p$ $r = \log_pq$ Group $PSL(2,q)$ Order (= $q^3 - q$ if $q$ even, $(q^3 - q)/2$ if $q$ odd) Automorphism group (equals $P\Gamma L(2,q)$) Order (= $r(q^3 - q)$) Outer automorphism group Order ( $r$ if $q$ even, $2r$ if $q$ odd) Number of endomorphisms (equals 1 + size of automorphism group for $q \ge 4$) Information on endomorphism structure
2 2 1 symmetric group:S3 6 symmetric group:S3 6 trivial group 1 10 endomorphism structure of symmetric group:S3
3 3 1 alternating group:A4 12 symmetric group:S4 24 cyclic group:Z2 2 33 endomorphism structure of alternating group:A4
4 2 2 alternating group:A5 60 symmetric group:S5 120 cyclic group:Z2 2 121 endomorphism structure of alternating group:A5
5 5 1 alternating group:A5 60 symmetric group:S5 120 cyclic group:Z2 2 121 endomorphism structure of alternating group:A5
7 7 1 projective special linear group:PSL(3,2) 168 projective general linear group:PGL(2,7) 336 cyclic group:Z2 2 337 endomorphism structure of projective special linear group:PSL(3,2)
8 2 3 projective special linear group:PSL(2,8) 504 Ree group:Ree(3) 1512 cyclic group:Z3 3 1513 endomorphism structure of projective special linear group:PSL(2,8)
9 3 2 alternating group:A6 360 automorphism group of alternating group:A6 1440 Klein four-group 4 1441 endomorphism structure of alternating group:A6