$p^{6}$阶群的Bogomolov乘子
Bogomolov multipliers of some groups of order $p^{6}$
摘要：
设$G$是有限群，$V$是$G$在复域$\mathbb{C}$上的忠实有限维表示，$\mathbb{C}(V)^{G}$是相应的不变域. $G$的Bogomolov乘子$B_{0}(G)$正则同构于非分歧上同调群$H_{\textrm{nr}}^{2}(\mathbb{C}(V)^{G},\mathbb{Q}/\mathbb{Z})$，Saltman（1984）和Bogomolov（1988）曾用它来为$\mathbb{C}$上的有限$p$ -群$P$的$\mathbb{C}(V)^{P}$的合理性问题提供反例。本文研究了$B_{0}(P)$的消失性质，其中$P$表示$p\geqslant 3$的$p^{6}$阶的$p$ -群.
英文摘要：
Let $G$ be a finite group, $V$ a faithful finite-dimensional representation of $G$ over the complex field $\mathbb{C}$ and $\mathbb{C}(V)^{G}$ be the corresponding invariant field. The Bogomolov multiplier $B_{0}(G)$ of $G$ is canonically isomorphic to the unramified cohomological group $H_{\textrm{nr}}^{2}(\mathbb{C}(V)^{G},\mathbb{Q}/\mathbb{Z})$, which has been used by Saltman (1984) and Bogomolov (1988) to provide counter-examples to the rationality problem of $\mathbb{C}(V)^{P}$ for finite $p$-groups $P$ over $\mathbb{C}$. In this paper, we investigate the vanishing property of $B_{0}(P)$, where $P$ denotes a $p$-group of order $p^{6}$ for $p\geqslant 3$.
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