LET X BE a space (i.e. simplicial set) with base point, which is nilpotent (i.e. connected and its Postnikov tower can be refined to a tower of principal fibrations) and on which a finite group T acts, keeping the base point fied. If R is a subring of the rationals, then the simplicial group GX; (the R-completion of the loop group GX) has (because X is nilpotent) thk homotopy type of the loops on the R-localization of X and its homotopy groups ?T,,GX~ = R @ n,,+,X (n > 0) are R[ T]-modules. Let C denote the center of T and assume, for the moment, that b-’ E R, where b is the order of T. Then the main purpose of this note is to split GX;, in a natural manner into a direct product of spaces with T-action, and even finer into a direct product of spaces with C-action. These splittings of GX; will realize the algebraic splittings of its homotopy groups as R[ T]-modules, provided by ordinary representation theory. Recall that, if b-* E R, where b is the order of T, then, according to ordinary representation theory, there is an essentially unique finest natural splitting of R[ T]-modules into a direct product of R[T]-modules with the following properties: (i) there are as many factors as T has conjugacy classes of cyclic subgroups, (ii) one of the factors (say the first) consists exactly of the elements fixed under T, and (iii) each of the other factors admits in turn an essentially unique finest natural splitting into a direct product of a finite number of isomorphic R[C]-modules, where C is the center of T. We show that this finest natural splitting of the homotopy groups of GXi into R[ T]-modules can be realized by a natural splitting of all of GX; into a direct product of spaces with T-action. In this splitting the analogues, (ii)’ and (iii)’ below, of (ii) and (iii) above, hold: (ii)’ The first factor is (GXi)T, the simplicial subgroup that is fixed under T, and ?*(GXi)’ = (T*GX~)~. By applying the classifying space functor @ one thus gets a space WGX; (with T-action), which has the homotopy type of the R-localization of X, and for which P*( WGX;)’ = (7r* wGXi)T. Thus GX; and WGXi have the property that “the homotopy of theirfixed point sets is the fixed point set of their homotopy”. (iii)’ Each of the other factors admits a further natural splitting into a direct product of a finite number of isomorphic spaces with C-action realizing the further splitting of (iii) of the other factors of the homotopy groups into R[C]-modules. If b-’ 6Z R, then most of this still goes through. One needs no restriction on R in order to realize any natural splitting of R[T]-modules into a direct product of R[ T] or R[C]-modules by means of a natural splitting of GX; into a direct product of spaces with C-action. However, one ,has to assume that d-’ E R, where d denotes the order of TIC, in order to realize any natural splitting of R[ T]-modules into a direct product of R[ T]-modules by means of a natural splitting of GX,^ into a direct product of spaces with T-action and not merely C-action.