Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called \emph{coregular} if the invariant ring is generated by algebraically independent homogeneous invariants, and the \emph{direct summand property} holds if there is a surjective $k[V]^G$-linear map $\pi\colon k[V]\to k[V]^G$. The following Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds.

An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.

Let $C_p$ denote the cyclic group of order $p$, where $p \geq 3$ is prime. We denote by $V_n$ the indecomposable $n$ dimensional representation of $C_p$ over a field $\mathbb F$ of characteristic $p$. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants $\mathbb F[V_2 \oplus V_2 \oplus V_3]^{C_p}$.

In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals.

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial $p$-group over a field of characteristic $p$, the ring of vector invariants of $m$ copies of that representation is not \comac\ for $m\geq 3$. In the second section of the paper we use Poincar\'e series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group $G$ and a faithful representation of dimension $n$ with $n>1$, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to $n(|G|-1)$. If the ring of invariants is a hypersurface, the upper bound can be improved to $|G|$.

Vector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, \eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB}, \cite{poly} and \cite{survey}. It is therefore all the more surprising that the {\it unpleasant} properties of these invariants may be derived from two unexpected, and remarkable, {\it nice} properties: namely for vector permutation invariants of the cyclic group $\mathbb{Z}/p$ of prime order in characteristic $p$ the image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon \mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal, and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im (\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern classes of the action.