# Endomorphism structure of 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: 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 $q$, the automorphism group of the special linear group of degree two $SL(2,q)$ 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 $PSL(2,q)$ $(q^3 - q)/\operatorname{gcd}(2,q-1)$ This is the quotient by the center. The center has order 1 and $PSL(2,q)$ has order $q^3 - q$ if $q$ is even. The center has order 2 and $PSL(2,q)$ has order $(q^3 - q)/2$ 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 $r$
$r\operatorname{gcd}(2,q-1)$
Case $q$ even: $r$
Case $q$ odd: $2r$

### Other endomorphisms

If $q$ 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 $q \ge 4$, the endomorphisms of $SL(2,q)$ are the automorphisms and the trivial endomorphism.

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