An exact sequence and triviality of Bogomolov multiplier of groups

TL;DR

This paper presents a new proof and exact sequence for the Bogomolov multiplier of finite groups, providing criteria for its triviality and classifying groups of order p^6 with trivial multiplier.

Contribution

It introduces a new proof of a Hopf-type formula, derives an exact sequence for the Bogomolov multiplier, and classifies certain groups with trivial multiplier.

Findings

New proof of Hopf-type formula for B_0(G)

Derived an exact sequence for the cohomological Bogomolov multiplier

Classified groups of order p^6 with trivial Bogomolov multiplier

Abstract

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of the cohomology classes which vanish after restricting to every abelian subgroup of $G$. We give a new proof of a Hopf-type formula for $B_0(G)$ and derive an exact sequence for the cohomological version of the Bogomolov multiplier. Using this exact sequence we provide necessary and sufficient conditions for the corresponding inflation homomorphism to be an epimorphism and to be the zero map. Finally, we give a complete list of groups of order $p^6$, for odd prime $p$, having trivial Bogomolov multiplier, so completing the 2020 investigation of Chen and Ma.