Help with computation of Scharlau invariant?

[Keith Dennis]

The problem is to compute something called the Scharlau invariant

(actually a generalization of the classical case) which determines

which groups act fix point freely in relatively prime characteristic.

The generalization characterizes when elements of minimal generating

sets act fix point free.

Let G be a non-trivial finite abelian p-group, R = Z[G]/I where I is

the generalized Scharlau ideal.  There are 2 interesting things to

find:

1.  The characteristic of the ring, called the generalized Scharlau

    invariant (this should be a power of p > 0)

and

2.  The structure of the ring, which should be a product of local

    rings I guess.

I know zilch about Macaulay2.  Any help or suggestions would be

greatly appreciated.

Keith

-- Problem:

-- Given a finite Abelian group, say p-group

-- G = ZZ/p^(a1) x ... x ZZ/p^(ak)

-- (where a1 <= a2 <= ... <= ak)

-- Consider the group ring of G:

-- R = ZZ[x_1..x_k]/(x_i^(p^ai)-1, all i)

-- and inside this group ring consider the ideal

-- I = <N(g) : g = (b1,...,bk), with some bi non-zero mod p>

   that is, g not the identity

-- and

-- N(g) = 1+g+g^2+...+g^((order of g) - 1)  [sum of group elements in

   cyclic subgroup generated by g]

-- Problem:  find the characteristic of the ring R/I.  Also find its structure.