Kaleidoscopical con gurations in G-spaces

Let G be a group and X be a G-space with the action G X ! X, (g; x) 7! gx. A subset F of X is called a kaleidoscopical con guration if there exists a coloring : X ! C such that the restriction of on each subset gF , g 2 G, is a bijection. We present a construction (called the splitting construction) of kaleidoscopical con gurations in an arbitrary G-space, reduce the problem of characterization of kaleidoscopical con gurations in a nite Abelian group G to a factorization of G into two subsets, and describe all kaleidoscopical con gurations in isometrically homogeneous ultrametric spaces with nite distance scale. Also we construct 2 (unsplittable) kaleidoscopical con gurations of cardinality c in the Euclidean space