On schemes originated from Ferrero pairs

Abstract

‎‎The Frobenius complement of a given Frobenius group acts on its kernel‎. ‎The scheme which is arisen from the orbitals of this action is called Ferrero pair scheme‎. ‎In this paper‎, ‎we show that the fibers of a Ferrero pair scheme consist of exactly one singleton fiber and every two fibers with more than one point have the same cardinality‎. ‎Moreover‎, ‎it is shown that the restriction of a Ferrero pair scheme on each fiber is isomorphic to a regular scheme‎. ‎Finally‎, ‎we prove that for any prime $p$‎, ‎there exists a Ferrero pair $p$-scheme‎, ‎and if $p> 2$‎, ‎then the Ferrero pair $p$-schemes of the same rank are all isomorphic‎.