Norm formulas for finite groups and induction from elementary abelian subgroups

الأصل · EN

It is known that the norm map NG for a finite group G acting on a ring R is surjective if and only if for every elementary abelian subgroup E of G the norm map Nₑ for E is surjective. Equivalently, there exists an element xG in R with NG(xG) = 1 if and only for every elementary abelian subgroup E there exists an element xₑ in R such that Nₑ(xₑ) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xₑ. In this paper we present a method to solve this problem for an arbitrary group G and an arbitrary group action on a ring.Using this method, we obtain a complete solution of the problem for the quaternion and the dihedral 2-groups,and for a group of order 27. We also show how to reduce the problem to the class of (almost) extraspecial p-groups.