We study Noether's problem over Q for meta-cyclic groups. This paper is an extension of the previous work [2] , which was concerned with the cyclic group C n of order n. We shall give a simple description of the action of the normalizer of C n in S n to the function field Q(x 1 , • • • , x n ), in terms of the generators of the fixed field of C n given in [2] . Using this, we settle Noether's problem for the dihedral group of order 2n (n ≤ 6) and the Frobenius group of order 20 with explicit

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... struction of independent generators of the fixed fields. We shall also reconstruct some simple one-parameter families of cyclic and dihedral polynomials.