Journal of Algebra 303 (2006), 677-706
Mathematics Subject Classification (2000): 16W22, 16U99, 20C10, 20D15, 20J05
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 in R with N_G(x_G) = 1 if and only if for every elementary abelian subgroup E there exists an element x_E in 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.
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(12 décembre 2006)