By Kondratev A.S.

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This completes the proof. 8 (Hilbert). If G is finite, then J is finitely generated as an algebra. See Flatto [98] for an account of Noether’s generalisation of this result. In the remainder of this section we present another approach based on knowing that S is an integral extension of J. Let A and B be commutative rings and suppose that A is a subring of B. 9. The extension A ⊆ B is of finite type if B is finitely generated as an A-algebra; it is finite if B is finitely generated as an A-module.

I r . Proof. Without loss of generality we may suppose that Ii = 0 for all i. We shall show by induction on the degree that every homogeneous polynomial P ∈ J is a (homogeneous) polynomial in I1 , I2 , . . , Ir . This is certainly true for polynomials of degree 0 and so we shall suppose that the degree of P is at least 1. In this case we have P ∈ J + ⊆ F and therefore we may write P = P1 I1 + P2 I2 + · · · + Pr Ir for some P1 , P2 , . . , Pr ∈ S. In addition, for all i, we may choose Pi to be homogeneous of degree deg P − deg Ii .

Thus µm G is represented as a subgroup of U (V ) and the subspaces Ce1 , Ce2 , . . , Cen form a system of imprimitivity for µm G. 6. With B := µnm as above and for each divisor p of m let A(m, p, n) := { (θ1 , θ2 , . . , θn ) ∈ B | (θ1 θ2 · · · θn )m /p = 1 }. It is immediate that A(m, p, n) is a subgroup of index p in B that is invariant under the action of Sym(n). The group G(m, p, n) is defined to be the semidirect product of A(m, p, n) by the symmetric group Sym(n). This notation was introduced by Shephard and Todd.