Bogomolov multipliers of some groups of order $p^6$

Abstract

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 $H_{\text{nr}}^2(\mathbb{C}(V{)}^G,\mathbb{Q}/\mathbb{Z})$, 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 $p⩾3$.

To appear in Comm. Algebra

References in corpus (3)

• Noether's problem for abelian extensions of cyclic $p$-groups
• Bogomolov multipliers for some $p$-groups of nilpotency class 2
• On $\CYRSH$-rigidity of groups of order $p^6$