Norm formulas for finite groups and induction from elementary abelian subgroups

Eli Aljadeff; Christian Kassel

doi:10.1016/j.jalgebra.2006.03.045

Norm formulas for finite groups and induction from elementary abelian subgroups

Eli Aljadeff, Christian Kassel^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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 ∈ R with N_G (x_G) = 1 if and only for every elementary abelian subgroup E there exists an element x_E ∈ 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.

Original language	English
Pages (from-to)	677-706
Number of pages	30
Journal	Journal of Algebra
Volume	303
Issue number	2
DOIs	https://doi.org/10.1016/j.jalgebra.2006.03.045
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

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10.1016/j.jalgebra.2006.03.045

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title = "Norm formulas for finite groups and induction from elementary abelian subgroups",

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.",

keywords = "Dihedral group, Extraspecial group, Group action, Group cohomology, Noncommutative ring, Norm map, Quaternion group, p-Group",

author = "Eli Aljadeff and Christian Kassel",

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doi = "10.1016/j.jalgebra.2006.03.045",

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T1 - Norm formulas for finite groups and induction from elementary abelian subgroups

AU - Aljadeff, Eli

AU - Kassel, Christian

PY - 2006/9/15

Y1 - 2006/9/15

N2 - 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.

AB - 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.

KW - Dihedral group

KW - Extraspecial group

KW - Group action

KW - Group cohomology

KW - Noncommutative ring

KW - Norm map

KW - Quaternion group

KW - p-Group

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DO - 10.1016/j.jalgebra.2006.03.045

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VL - 303

SP - 677

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JO - Journal of Algebra

JF - Journal of Algebra

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ER -

Norm formulas for finite groups and induction from elementary abelian subgroups

Abstract

Keywords

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