By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fai...

Let K be any field, G be a finite group. Let G act on the rational function field K(x(g) : g ∈ G) by K-automorphisms and h · x(g) = x(hg). Denote by K(G) = K(x(g) : g ∈ G) the fixed field. Noether’s problem asks whether K(G) is rational (= purely transcendental) over K. We will give several reduction theorems for solving Noether’s problem. For example, let G̃ = G × H be a direct product of finit...

Our aim is to study the set of K-rational expressions describing rational series. More precisely we are concerned with the definition of quotients of this set by coarser and coarser congruences which lead to an extension – in the case of multiplicities – of some classical results stated in the Boolean case. In particular, multiplicity analogues of the well known theorems of Brzozowski and Antim...

We prove an extension of the following result of Lubotzky and Madid on the rational cohomology of a nilpotent group G: If b1 < ∞ and G⊗Q = 0,Q,Q then b2 > b1/4. Here the bi are the rational Betti numbers of G and G ⊗ Q denotes the Malcev-completion of G. In the extension, the bound is improved when the relations of G are known to all have at least a certain commutator length. As an application ...