Symmetries of the Julia sets of König's methods for polynomials

Let Σ(R)Σ(R) be the group of symmetries of the Julia set of a rational map R   and Kf,nKf,n be the König's method for polynomial f   of order n(≥2). For any given integer n≥2n≥2, we prove that if f   is in normal form, then Σ(f)Σ(f) is a subgroup of Σ(Kf,n)Σ(Kf,n). We also obtain a necessary and sufficient condition for the Julia set of Kf,nKf,n to be a line.