Endomorphism structure of projective special linear group of degree three over a finite field
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
Let where is the underlying prime. The information is presented below:
|automorphism group||projective outer semilinear group of degree three||When , this is the same as the projective outer linear group of degree three.|
|inner automorphism group||projective special linear group of degree three||The group is identified with its inner automorphism group because it is a centerless group.|
|outer automorphism group||
Case is not 1 mod 3: direct product of cyclic group of order 2 and cyclic group of order
Case is 1 mod 3: Direct product of cyclic group of order 6 and cyclic group of order
Case is not 1 mod 3:
Case is 1 mod 3:
Projective special linear group is simple, and in particular is always simple, so the only endomorphisms are the automorphisms and the trivial endomorphism.