Torsion matrices over commutative integral group rings

Let ZA he the integral group ring of a finite abelian group A, and n a positive integer greater than 5. We provide conditions on n and A under under which every torsion matrix U, with identity augmentation, in GL(n)(ZA) is conjugate in GL(n)(QA) to a diagonal matrix with group elements on the diagonal. When A is infinite, we show that under similar conditions, U has a group trace and is stably conjugate to such a diagonal matrix.