On the sharply 1-transitive subsets of \(PGL(2,p^ m)\) (Q1100434)

The main aim of the author's investigation is to prove a proposition which can be considered as one further step towards the verification of the following conjecture: If p is odd then every sharply 1-transitive subset R of \(PGL(2,p^ m)\) such that id\(\in R\) is necessarily a subgroup. The author enumerates some low values of \(p^ m\) for which the conjecture was verified to be true, and then he proves the following main result: If p is odd and R is a sharply 1-transitive subset of \(PGL(2,p^ m)\) such that id\(\in R\), then either \(<R>=R\) or \(<R>=PSL(2,p^ m)\) or \(<R>=PGL(2,p^ m)\).