Norm formulas for finite groups and induction from elementary abelian subgroups

Details Norm formulas for finite groups and induction from elementary abelian subgroups Norm formulas for finite groups and induction from elementary abelian subgroups

2004-02-09

arxiv.org/abs/math/0402145v2

Journal of Algebra 303 (2006), 677-706

Mathematics

Rings and Algebras

31 pages. In Section 1 a universal ring and the proof of the existence of formulas for any finite group were added

Scientific paper

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 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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