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