Explicit norm one elements for ring actions of finite Abelian groups

Eli Aljadeff; Christian Kassel

doi:10.1007/BF02773156

Explicit norm one elements for ring actions of finite Abelian groups

Eli Aljadeff, Christian Kassel

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

3 Scopus citations

Abstract

It is known that the norm map N_G for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map N_U is surjective. Equivalently, there exists an element x_G ∈ R satisfying N_G(x_G) = 1 if and only if for every elementary abelian subgroup U there exists an element x_U ∈ R such that N_U(x_U) = 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_U. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.

Original language	English
Article number	BF02773156
Pages (from-to)	99-108
Number of pages	10
Journal	Israel Journal of Mathematics
Volume	129
DOIs	https://doi.org/10.1007/BF02773156
State	Published - 2002
Externally published	Yes

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title = "Explicit norm one elements for ring actions of finite Abelian groups",

abstract = "It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map NU is surjective. Equivalently, there exists an element xG ∈ R satisfying NG(xG) = 1 if and only if for every elementary abelian subgroup U there exists an element xU ∈ R such that NU(xU) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xU. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.",

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AU - Aljadeff, Eli

AU - Kassel, Christian

PY - 2002

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N2 - It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map NU is surjective. Equivalently, there exists an element xG ∈ R satisfying NG(xG) = 1 if and only if for every elementary abelian subgroup U there exists an element xU ∈ R such that NU(xU) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xU. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.

AB - It is known that the norm map NG for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map NU is surjective. Equivalently, there exists an element xG ∈ R satisfying NG(xG) = 1 if and only if for every elementary abelian subgroup U there exists an element xU ∈ R such that NU(xU) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for xG in terms of the elements xU. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.

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VL - 129

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JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

M1 - BF02773156

ER -

Explicit norm one elements for ring actions of finite Abelian groups

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