Milnor number of an invariant singularity: generalization of Chulkov's inequality

We prove a lower bound for the Milnor number of function germ invariant with respect to a finite abelian group action. It is shown that this bound is tight for functions of arbitrarily many variables. We also prove the function germs that reach this lower bound are equivariantly stable, i.e. invariant analogues of Morse singularities.