Radó, F. Israel J. Math. (1986) 53: 217. doi:10.1007/BF02772860
LetV be a metric vector space over a fieldK, dimV=n<∞, and let δ:V×V→K denote the corresponding distance function. Given a mappingσ:V→V such that δ(p,q) = 1⇒ δ(pσ,qς) = 1, ifn=2, indV=1 and charK≠2, 3, 5, thenσ is semilinear , ; ifn≧3,K=R and the distance function is either Euclidean or Minkowskian, thenσ is linear , . Here the following is proved: IfK=GF(pm),p>2 andn≧3, thenσ is semilinear (up to a translation), providedn≠0, −1, −2 (modp) or the discriminant ofV satisfies a certain condition. The proof is based on the condition for a regular simplex to exist in a Galois space, which may be of interest for its own sake.