Endomorphism structure of 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: 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.

Facts about endomorphism structure