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

This article gives specific information, namely, endomorphism structure, about a family of groups, namely: projective special linear group of degree two.
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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 $p$	$r=\log _{p}q$	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 $q\geq 4$)	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