  We consider a new subclass of quadratic stochastic (evolutionary) operators
on the simplex indexed by a finite Abelian group G with heredity law \mu. With
the help of the notion of s(\mu)-invariant subgroups, where s(\mu) denotes the
support of \mu in G, we prove that almost all (w.r.t.\ Lebesgue measure)
trajectories of such operators converge to a unique fixed point which is the
center of the simplex. We also identify and describe the periodic trajectories
of the operator and give conditions for regularity and periodicity.
