PGL(2,Fq) acting on Fq(x)
Xiang‐dong Hou · Communications in Algebra · 2019
Let Fq(x) be the field of rational functions over Fq and treat PGL(2,Fq) as the group of degree one rational functions in Fq(x) equipped with composition. PGL(2,Fq) acts on Fq(x) from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup H of PGL(2,Fq) is the stabilizer of some rational function πH(x)∈Fq(x) with deg πH=|H| under this action, where πH(x) is uniquely determined by H up to a left composition by an element of PGL(2,Fq). In this article, we determine the rational function πH(x) explicitly for every H