Endomorphism structure of special linear group of degree two over a finite field
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
Let where is the underlying prime. The information is presented below:
|automorphism group||projective semilinear group of degree two||When , i.e., the field is a prime field, then the automorphism group is just .|
|inner automorphism group||projective special linear group of degree two||This is the quotient by the center. The center has order 1 and has order if is even. The center has order 2 and has order if is odd.|
|outer automorphism group|| Case even: cyclic group of order
Case odd: Direct product of cyclic group of order 2 and cyclic group of order
If 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 , the endomorphisms of are the automorphisms and the trivial endomorphism.
The cases (giving symmetric group:S3 -- see endomorphism structure of symmetric group:S3) or (giving special linear group:SL(2,3) -- see endomorphism structure of special linear group:SL(2,3)) are somewhat different.