Bogomolov multipliers and retract rationality for semi-direct products

TL;DR

This paper investigates the properties of Bogomolov multipliers for finite groups, their behavior under group products and semi-direct products, and explores conditions under which certain group actions are retract rational, extending previous results.

Contribution

It establishes new isomorphism results for Bogomolov multipliers under direct and semi-direct products, and generalizes conditions for retract rationality of group actions over fields.

Findings

B0(G1×G2) ≅ B0(G1) × B0(G2) for finite groups G1, G2.

B0(G) ≅ B0(N)^{G0} × B0(G0) for G=N⋉G0 with coprime orders.

Existence of non-direct-product p-groups with large subgroups in their Bogomolov multipliers.

Abstract

Let $G$ be a finite group. The Bogomolov multiplier $B_0(G)$ is constructed as an obstruction to the rationality of $\bm{C}(V)^G$ where $G\to GL(V)$ is a faithful representation over $\bm{C}$. We prove that, for any finite groups $G_1$ and $G_2$, $B_0(G_1\times G_2)\xrightarrow{\sim} B_0(G_1)\times B_0(G_2)$ under the restriction map. If $G=N\rtimes G_0$ with $\gcd\{|N|,|G_0|\}=1$, then $B_0(G)\xrightarrow{\sim} B_0(N)^{G_0}\times B_0(G_0)$ under the restriction map. For any integer $n$, we show that there are non-direct-product $p$-groups $G_1$ and $G_2$ such that $B_0(G_1)$ and $B_0(G_2)$ contain subgroups isomorphic to $(\bm{Z}/p \bm{Z})^n$ and $\bm{Z}/p^n \bm{Z}$ respectively. On the other hand, if $k$ is an infinite field and $G=N\rtimes G_0$ where $N$ is an abelian normal subgroup of exponent $e$ satisfying that $\zeta_e\in k$, we will prove that, if $k(G_0)$ is retract $k$-rational, then $k(G)$ is also retract $k$-rational provided that certain "local" conditions are satisfied; this result generalizes two previous results of Saltman and Jambor \cite{Ja}.