Norm formulas for finite groups and induction from elementary abelian subgroups

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 NE for E is surjective. Equivalently, there exists an element X-G is an element of R with N-G (x(G)) = I if and only for every elementary abelian subgroup E there exists an element x(E) is an element of 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. (c) 2006 Elsevier Inc. All rights reserved.