Endomorphism structure of projective special linear group of degree two over a finite field
This article gives specific information, namely, endomorphism structure, about a family of groups, namely: projective special linear group of degree two.
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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||The group is identified with its inner automorphism group because it is a centerless group. The order is if is even, and 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
| Case even:
If is 4 or more, the group PSL(2,q) is simple, so the only endomorphisms are the trivial endomorphism and the automorphisms. If (giving symmetric group:S3) or (giving alternating group:A4) then there are other endomorphisms with nontrivial kernels.