On representations of permutations groups as isometry groups of \(n\)-semimetric spaces

Abstract

We prove that  every finite  permutation group can be represented as the isometry group of some \(n\)-semimetric space.  We show that if a finite  permutation group can be realized as the isometry group of some \(n\)-semimetric space then this permutation group can be represented as the isometry group of some \((n+1)\)-semimetric space. The notion of the semimetric rank of a permutation group is   introduced.