Abstract
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 ∈ R with NG (xG) = 1 if and only for every elementary abelian subgroup E there exists an element xE ∈ 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.
Original language | English |
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Pages (from-to) | 677-706 |
Number of pages | 30 |
Journal | Journal of Algebra |
Volume | 303 |
Issue number | 2 |
DOIs | |
State | Published - 15 Sep 2006 |
Externally published | Yes |
Keywords
- Dihedral group
- Extraspecial group
- Group action
- Group cohomology
- Noncommutative ring
- Norm map
- Quaternion group
- p-Group