##
Definitions and Notation

**Noether Numbers:**
Suppose that V is a representation of an algebraic group, G, over a
field, **k**. Suppose that the ring of invariants
**k**[V]^{G} is finitely generated. This is guarenteed
to happen if G is linearly reductive or if G is finite, for example.
Let f_{1},f_{2},...,f_{r} be a homogeneous
minimal set of algebra generators for the algebra
**k**[V]^{G}.
The number max{deg(f_{i}) | i=1,2,...,r} is called the
*Noether Number *of the representation and is denoted by
b(**k**[V]^{G}) and by
b(V,G).
More generally if R is any finitely generated graded algebra we
denote by b(R) the largest degree of a
generator in a homogeneous minimal generating set. More generally
still, if I is a homogeneous ideal in a Noetherian algebra we
denote by b(I) the largest degree
of a generator in a homogeneous minimal generating set for I.
If G is a finite group then we have two possibilities: either the
representation is *modular *or it is *non-modular*.
If the characteristic of **k** is a positive prime, p, and if p
divides the order of G then we say the representation is modular;
otherwise it is non-modular. In 1916 Emmy Noether proved that if
the characteristic of **k** is zero then
b(V,G) <= |G|. Recently,
P. Fleishmann and J. Forgarty independently proved that for any
non-modular representation
b(V,G) <= |G|. It is well known that
b(V,G) may exceed |G| for modular
representations.

If G is a finite group, we define the *Transfer* (also
called *Trace*) *homomorphism*,
Tr^{G}: **k**[V] --> **k**[V]^{G} by
Tr^{G}(f):=S
_{g e G} (gf).
Similarly we define the norm of a variable *x* as
N(*x*):=P
_{g e G} (g*x*).
For non-modular representations, the map Tr^{G} is
surjective; for modular representations it is not onto
and its image, Im Tr^{G}, is a non-zero proper ideal
in **k**[V]^{G}.
Related to the transfer is the notion of *orbit
sums*. Given an element f of **k**[V] we consider its
orbit under the action of G:
{gf | g e G} =: {f=f_{1}, f_{2}, ..., f_{t}}.
The orbit sum of f is the element
*O*(f) := f_{1} + f_{2} + ... + f_{t}.
Clearly *O*(f) is an invariant. Moreover Tr^{G}(f)
is an positive integer multiple of *O*(f).
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