## Abstract

Let k be any field, G be a finite group acting on the rational function field k(x_{g} : g ε G) by h · x_{g} = x_{hg} for any h, g ε G. Define k(G) = k(x_{g} : g ε G)^{G}. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if ℂ(G) is rational over ℂ, then B_{0}(G) = 0 where B_{0}(G) is the unramified Brauer group of ℂ(G) over ℂ. Bogomolov showed that, if G is a p-group of order p5, then B_{0}(G) = 0. This result was disproved by Moravec for p = 3, 5, 7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p5. Then B_{0}(G) ≠ 0 if and only if G belongs to the isoclinism family π_{10} in R. James's classification of groups of order p^{5}.

Original language | English |
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Pages (from-to) | 689-714 |

Number of pages | 26 |

Journal | Asian Journal of Mathematics |

Volume | 17 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

## Keywords

- Bogomolov multipliers
- Noether's problem
- Rationality
- Rationality problem
- Retract rationality
- Unramified Brauer groups