## 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 |
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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
- p-Group
- Quaternion group