Finite -arc transitive Cayley graphs and flag-transitive projective planes

In this paper, a characterisation is given of finite -arc transitive Cayley graphs with . In particular, it is shown that, for any given integer with and , there exists a finite set (maybe empty) of -transitive Cayley graphs with such that all -transitive Cayley graphs of valency are their normal covers. This indicates that -arc transitive Cayley graphs with are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.