Let denote the semigroup, under composition, of all entire functions of a complex variable and for , let Γ(f) denote the Schützenberger group of the ℋ-class of which contains f. The main results of this paper completely describe Γ(P) where P is a polynomial function. Γ(P) is trivial if P is a constant function. If Deg (P) (the degree of P) is one, then Γ(P) is isomorphic to the group of all matrices of the form where a and b are complex numbers with a ≠ 0. If Deg (P) > 1, then Γ(P) is isomorphic to ℂM, the multiplicative group of nonzero complex numbers if and only if P(z) = α(z +β)n + γ where α ≠ 0. If P is not of this form and Deg (P) > 1, then Γ (P)is a finite cyclic group and its order can be determined by defining (where αn−1 and α n are the coefficients in P of zn−1 and zn respectively) and then inspecting the coefficients of Q.