On representations of permutations groups as isometry groups of n-semimetric spaces

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