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

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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