Norm formulas for finite groups and induction from elementary abelian subgroups

Abstract

It is known that the norm map N G 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 N E for E is surjective. Equivalently, there exists an element x G ∈ R with N G ( x G ) = 1 if and only for every elementary abelian subgroup E there exists an element x E ∈ R such that N E ( x E ) = 1 . When the ring R is noncommutative, it is an open problem to find an explicit formula for x G in terms of the elements x E . 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.