Endomorphism structure of projective special linear group of degree three 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 three.
View endomorphism structure of group families | View other specific information about projective special linear group of degree three

Automorphism structure

For any prime power q, the automorphism group of the projective special linear group of degree three PSL(3,q) over the finite field \mathbb{F}_q is the projective outer semilinear group of degree three PO\Gamma L(3,q).

Let q = p^r where p is the underlying prime. The information is presented below:

Construct Value Order Comment
automorphism group projective outer semilinear group of degree three PO\Gamma L(3,q) 2rq^3(q^3 - 1)(q^2 - 1) When r = 1, this is the same as the projective outer linear group of degree three.
inner automorphism group projective special linear group of degree three PSL(3,q) q^3(q^3 - 1)(q^2 - 1)/\operatorname{gcd}(3,q-1) The group is identified with its inner automorphism group because it is a centerless group.
outer automorphism group
Case q is not 1 mod 3: direct product of cyclic group of order 2 and cyclic group of order r
Case q is 1 mod 3: Direct product of cyclic group of order 6 and cyclic group of order r
2r \operatorname{gcd}(3, q - 1)
Case q is not 1 mod 3: 2r
Case q is 1 mod 3: 6r

Other endomorphisms

Projective special linear group is simple, and in particular PSL(3,q) is always simple, so the only endomorphisms are the automorphisms and the trivial endomorphism.