Endomorphism structure of projective general linear group of degree two over a finite field
This article gives specific information, namely, endomorphism structure, about a family of groups, namely: projective general linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.
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|automorphism group||projective semilinear group of degree two||Semidirect product where is cyclic and generated by the Frobenius .|
|inner automorphism group||itself||The group is a centerless group hence equals its inner automorphism group.|
|outer automorphism group||The outer automorphism group is trivial, and hence, the group is a complete group (note that it's already centerless) in the case , i.e., in the case that . In other words, a projective general linear group of degree two over a finite field is complete if and only if that field is a prime field.|