##
Some Conjectures of Mine and Others

These conjectures are all original but I make
no claims of priority.

The conjectures concern modular invariant theory and many of them
concern
the Noether number. They are expressed using standard notation
and definitions.

**False Conjecture 1:** Let G be a finite group.
Let R denote **k**[V]^{G}
and let I denote the image of the transfer homomorphism. I had conjectured
that b(R/I) <=
|G| but Peter Fleischmann, Gregor Kemper and Jim Shank have just recently
produced a counter example to this.
**Conjecture 2:** Let G be a finite p-group where **k** is of
characteristic p > 0.
If Im Tr^{G} is a principal ideal
then **k**[V]^{G} is a polynomial ring.
Jim Shank and I proved the
converse of this conjecture when **k** is the prime field
**F**_{p}=GF(p).
Bram Broer has recently proved the converse of the conjecture for any
field **k** of characteristic p.
**No 3's Conjecture:** Let V_{p} denote the regular
representation of the cyclic group Z/p over the finite field of
prime order **F**_{p}. Jim Shank and I had conjectured
that
b(**F**_{p}[V_{p}]^{Z/p}) = 2p-3.
We had proven that b(**F**_{p}[V_{p}]^{Z/p})
>= 2p-3.
In a recent preprint (Fall 2005) P. Fleischmann, M. Sezer, R.J. Shank and C.F. Woodcock
prove that b(**F**_{p}[V_{p}]^{Z/p})
is indeed 2p-3.
I conjecture that
**F**_{p}[V_{p}]^{Z/p}
is generated by Norms together with the orbit sums of the monomials (with respect to the usual
permutation basis) which have no
exponent exceeding 2.
**Conjecture 4** If V is a representation of G and W is a subrepresentation of V then
b(**k**[W]^{G}) <= b(**k**[V]^{G}).
Jim Shank and I have proven this conjecture for G=Z/p.
**Conjecture 5** Let V be a representation of a p-group with a polynomial ring of invariants.
I conjecture that it is always possible to take a (non-lnear) norm of a linear form as one of
the minimal generators for the ring of invariants. I even think this is true without the
assumption that the ring of invariants is polynomial. This more general form is true for cyclic
p-groups.

Comments,
proofs (or disproofs) are welcome!

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