Equivariant embeddings of manifolds into Euclidean spaces

Suppose a finite group G acts on manifold M. By theorem of Mostow, also Palais, there is G-equivariant embedding M into the m-dimensional Euclidean space Rm for some m. We are interested in explicit bounds such First we provide an upper bound: exists Rd|G|+1, where |G| order and embeds Rd. Next lower bound cyclic action G: If l points having pairwise co-prime lengths G-orbits greater than 1 Rm, then m≥2l. Some applications to surfaces given.