Bogomolov multipliers for some $p$-groups of nilpotency class 2

TL;DR

This paper investigates the Bogomolov multiplier for certain p-groups of nilpotency class 2, providing conditions for its triviality and applying results to solve Noether's problem for specific groups.

Contribution

It establishes new criteria for the triviality of Bogomolov multipliers in p-groups of class 2 and confirms Noether's problem for specific 2- and 4-generator p-groups.

Findings

Triviality of B_0(G) is preserved under certain central product constructions.

Confirmed Noether's problem for all 2-generator p-groups of nilpotency class 2.

Solved Noether's problem for a series of 4-generator p-groups of nilpotency class 2.

Abstract

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of $G$. The triviality of the Bogomolov multiplier is an obstruction to Noether's problem. We show that if $G$ is a central product of $G_1$ and $G_2$, regarding $K_i\leq Z(G_i), i=1,2$, and $\theta:G_1\to G_2$ is a group homomorphism such that its restriction $\theta\vert_{K_1}:K_1\to K_2$ is an isomorphism, then the triviality of $B_0(G_1/K_1), B_0(G_1)$ and $B_0(G_2)$ implies the triviality of $B_0(G)$. We give a positive answer to Noether's problem for all $2$-generator $p$-groups of nilpotency class $2$, and for one series of $4$-generator $p$-groups of nilpotency class $2$ (with the usual requirement for the roots of unity).