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Let $\rho : G \hra \GL(n,\ \F)$ be a faithful representation of the finite group
$G$ over the field $\F$\/. In 1916 E. Noether proved that
for $\F$ of characteristic zero the ring of
invariants $\F[V]^G$ is generated as an algebra by the invariant polynomials of
degree at most $|G|$\/. This result has been generalized to the case
where the characteristic of $\F$ is
greater than $|G|$\/, or when the characteristic of $\F$ is prime to the
order of $G$ and the group $G$ is solvable.
The result for fields whose characteristic is greater than the order of the
group follows directly from the fact that in this case the ring of invariants
is generated by orbit Chern classes. However this can fail in the more general
nonmodular situation; e.g., for the quaternion subgroup of $\GL(2, \F_3).
In this note we show how to rework Noether's proof to
yield a more more refined notion of orbit Chern classes (here called the
{\bf fine} orbit Chern classes). This leads to a very general
result where Noether's bound holds, that includes both the previously mentioned
theorems, and in particular, shows that Noether's bound holds for the
alternating groups in the nonmodular case. We also illustrate with the example
of the quaternion group that these new, fine, orbit Chern classes are
sufficient to generate the ring of invariants in cases where orbit Chern classes
fail to do so.
An interesting group representational problem that arises in this connection is
the following. If $G$ is a finite group of order $n$ and
$\reg_G : G \longrightarrow \Sigma_n$ its' regular representation, then the
image of $G$ in $\Sigma_n$ is not {\it arbitrary}\/, in the sense that there is
always a proper subgroup of $\Sigma_n$\/, depending only on the orders of the
groups in a chain of maximal subgroups of $G$\/, that contains the image. Is
there a {\it smallest } such subgroup? If so how does it depend on $G$? What
is its structure? This note contains a very crude solution which is an iterated
wreath product of symmetric groups. This factorization of $\reg_G$
certainly has applications to the study of the Chern classes of the regular
representation of $G$\/.