Automorphisms of order 3 and 7 over a genus 3 curve (Q2367288)

This paper proves that if an algebraic curve of genus 3 over a field of characteristic \(p \neq 2,3,7\) possesses an automorphism of order 3 and one of order 7, then the automorphism group is simple of order 168. The proof depends on the Riemann Hurwitz formula (justified in characteristic \(p\) by a result of Roquette) and some elementary group theory. It seems to the reviewer that for non-hyperelliptic curves an argument on plane quartics shows that the existence of an automorphism of order 7 already suffices to characterize Klein's curve in characteristic \(\neq 7\) (while there is only one hyperelliptic curve with such an automorphism).