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 TrG 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 Fp=GF(p). Bram Broer has recently proved the converse of the conjecture for any field k of characteristic p.
No 3's Conjecture: Let Vp denote the regular representation of the cyclic group Z/p over the finite field of prime order Fp. Jim Shank and I had conjectured that b(Fp[Vp]Z/p) = 2p-3. We had proven that b(Fp[Vp]Z/p) >= 2p-3. In a recent preprint (Fall 2005) P. Fleischmann, M. Sezer, R.J. Shank and C.F. Woodcock prove that b(Fp[Vp]Z/p) is indeed 2p-3. I conjecture that Fp[Vp]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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