Let (G, *) be a finite group. 1. Then (say by left multiplication) G acts as a group LG of permutations on itself: LG := {L_g | g in G} is a subgroup of S(G), where L_g(x) := gx for all x in G and all g in G, and S(G) is the group of all permutations of the set G. 2. Definition: ----------- A left-invariant metric D on G is one such that for every g in G, the permutation L_g: G -> G is an isometry. 3. There is always a left-invariant metric D on G. For, pick any metric d on G, and average it over the action of LG: D(x,y) := (1/|G|) Sum_{g in G} d(L_g(x), L_g(y)) Then it readily follows that D(L_g(x), L_g(y)) = D(x,y) for every g in G. 4. Question: --------- Which finite groups G have some left-invariant metric D whose full group of isometries is precisely the left multiplications Isom((G,D)) = {L_g | g in G} ??? 5. Example: -------- Let G be the alternating group A_3. Every left-invariant metric D makes G into an equilateral triangle. But every equilateral triangle has a 6-element isometry group, S_3. —Dan