Norm formulas for finite groups and induction from elementary abelian subgroups

Eli Aljadeff, Christian Kassel*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 languageEnglish
Pages (from-to)677-706
Number of pages30
JournalJournal of Algebra
Volume303
Issue number2
DOIs
StatePublished - 15 Sep 2006
Externally publishedYes

Keywords

  • Dihedral group
  • Extraspecial group
  • Group action
  • Group cohomology
  • Noncommutative ring
  • Norm map
  • p-Group
  • Quaternion group

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