Existence and maximal corank of simple $Z_p$-invariant germs

In this paper we improve the previously achieved upper bound on the corank of an equivariantly stable singularity for a group of prime order. We also prove that the maximal corank of a simple ${\mathbb{Z}}_p$-invariant germ tends to infinity as $p$ increases and is asymptotically logarithmic, so the previously obtained bound is valid up to order of magnitude.