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This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better understanding of the multiplicative ideal theory of invariant rings, and connections between the Noether number and the Davenport constants of finite groups.

A quantum deformation of the classical conjugation action of GL(N, C) on the space of N × N matrices M(N, C) is defined via a coaction of the quantum general linear group O(GL q (N, C)) on the algebra of quantum matrices O(M q (N, C)). The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of O(M q (N, C)) generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction.

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For the finite groups with a cyclic subgroup of index two the exact degree bound for the generators of rings of polynomial invariants is determined.

The finite groups having an indecomposable polynomial invariant of degree at least half the order of the group are classified. It turns out that -apart from four sporadic exceptions-these are exactly the groups with a cyclic subgroup of index at most two. * The paper is based on results from the PhD thesis of the first author written at the Central European University.† The second author is partially supported by OTKA NK81203 and K101515.

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: If two points in the direct sum of the G-modules W and m copies of V can be separated by polynomial invariants, then they can be separated by invariants depending only on 2 dim(V ) variables of type V ; when G is reductive, invariants depending only on dim(V ) + 1 variables suffice. A similar result is valid for rational invariants. Explicit bounds on the number of type V variables in a complete system of typical separating invariants are given for the binary polyhedral groups, and this is applied to the invariant theory of binary forms.

Known results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given. MSC:13A50, 11B75, 13A02