Modular vector invariants

Abstract

Vector invariants of finite groups (see the introduction for definitions) provides, in general, counterexamples for many properties of the invariant theory when the characteristic of the ground field divides the group order. Noether number is such property. In this thesis, we improve a lower bound for Noether number given by Richman in 1996: namely, we give a lower bound depending on the Jordan canonical form of an element of order equal to characteristic of the field. This method yields an effective bound by means of simple arithmetic arguments. The results are valid for any faithful representation of the group, including reducible and irreducible ones. Also they are extended to any algebraic field extensions provided the characteristic of the field divides the group order.