Some Metrical Problems on Sn**Ce trvail a été effectué pendant le séjour des auteurs au Centre de Recherche de Mathématiques, Appliquées, Université de Mntréal.

We present some problems related to right- (or bi-)invariant metrics d on the symmetric group of permutations S n . Characterizations and constructions of bi-invariant extremal (in the corresponding convex cone) metrics are given, esp. for n ⩽ 5. We also consider special subspaces of the metric space (S n , d): unit balls, sets with prescribed distances ( L -cliques), “hamiltonian” sets. Here we give (for proofs, see [3]) some results and problems arising by analogy with extremal set systems. Related problems of coding type (with Hamming metric) are considered in [1, 4].