Abstract
It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map Nu is surjective. Equivalently, there exists an element XG G R satisfying NG (XG) = 1 if and only if for every elementary abelian subgroup U there exists an element xu E R such that Nu(xu) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for XG in terms of the elements xu. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.
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