Endomorphism structure of special linear group of degree two over a finite field

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 $q\geq 4$ , 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.

Facts about endomorphism structure