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.