Invariant metrics on finite groups

We study invariant and bi-invariant metrics on groups focusing finite G . show that non-equivalent (bi) are in 1-1 correspondence with unitary symmetric (conjugate) partitions To every metric group ( , d ) we associate to it the symmetry weighted graph of distances. Using these objects can classify all equivalence classes for small groups. then number give an expression such terms Bell numbers, closed expressions certain as abelian, dihedral, quasidihedral dicyclic characterize (finite or not) which is also bi-invariant. order up 32.