The labeling of the Hamming Space (Z(2)(2),d(h)) by the rotation group Z(4) and its coordinate-wise extension to Z(2)(2n) give rise to the concept of Z(4)-linearity. Attempts to extend this concept have been done in different ways. We deal with a natural extension question: Is there any pattern of a cyclic group G labeling of Z(m)(n), with the Hamming or Lee metric? The answer is no. Actually, we show here that Lee spaces do not allow even labelings by abelian groups, what lead us to construct labelings by semi-direct products of abelian groups. Labelings of general Hamming spaces and of Reed-Muller codes RM(1,m) are characterized here in the context of isometry groups. (C) 2002 Elsevier Science B.V. All rights reserved.26041699119136