Abstract. Let G be any finite group, G → GL(V ) be a representation of G, where V is a finite-dimensional vector space over an algebraically closed field k. Theorem. Assume that either char k = 0 or char k = p > 0 with p |G|. Then the quotient variety P(V )/G is projectively normal with respect to the line bundle L, where L is the descent of O(1) ⊗m from P(V ) with m = |G|!. This partially solves a question raised in the paper of Kannan, Pattanayak and Sardar [Proc.
Abstract. Let K be any field and G be a finite group. Let G act on the rational function fieldNoether's problem asks whether K(G) is rational (= purely transcendental) over K. We shall prove that K(G) is rational over K if G is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.
field K (G) will be rational (= purely transcendental) over K .Theorem. Let G be a finite group of order 32 with exponent e. If char K = 2 or K is any field containing a primitive eth root of unity, then K (G) is rational over K .
Denote by ed K (G) the essential dimension of G over K. If K is an algebraically closed field with char K = 0, Buhler and Reichstein determine explicitly all finite groups G with ed K (G) = 1 [Compositio Math. 106 (1997), Theorem 6.2]. We will prove a generalization of this theorem when K is an arbitrary field.
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