2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97823We show that the symmetry groups of the cut cone Cut(n) and the metric cone Met(n) both consist of the isometries induced by the permutations on {1,...,n}; that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4). This is then extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone.8 pages, LaTeX, 2 postscript figuresMetric GeometryGroup Theory52B15 (Primary); 51M20; 51F99; 20F29 (Secondary)text