Finite p-Irregular Subgroups of PGL(2,k)

In the late 19th century, Klein inaugurated a program for describing the finite subgroups of $PG{L}_2(k)$ by treating the case in which the field $k$ is the complex numbers. Gierster and Moore extended Klein’s arguments to deal with finite fields. In the past century, additional contributions to this problem were made by Serre, Suzuki, and Beauville, among others. We complete this program by giving a classification of the finite subgroups of $PG{L}_2(k)$ with order divisible by $p$, up to conjugation, for an arbitrary field $k$ of positive characteristic $p$.