Universal commutator relations, Bogomolov multipliers, and commuting probability

TL;DR

This paper establishes a threshold for the commuting probability in finite p-groups that guarantees the triviality of the unramified Brauer group, linking group structure to algebraic invariants and providing classifications and counterexamples.

Contribution

It introduces a sharp bound on commuting probability ensuring trivial Bogomolov multipliers, and characterizes groups with nontrivial multipliers, advancing understanding of group relations and algebraic invariants.

Findings

Bound on commuting probability for trivial unramified Brauer group

Classification of p-groups with nontrivial Bogomolov multipliers

Counterexamples to Bogomolov's claims on nilpotency class

Abstract

Let $G$ be a finite $p$-group. We prove that whenever the commuting probability of $G$ is greater than $(2p^2 + p - 2)/p^5$, the unramified Brauer group of the field of $G$-invariant functions is trivial. Equivalently, all relations between commutators in $G$ are consequences of some universal ones. The bound is best possible, and gives a global lower bound of $1/4$ for all finite groups. The result is attained by describing the structure of groups whose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all of their proper subgroups and quotients are trivial. Applications include a classification of $p$-groups of minimal order that have nontrivial Bogomolov multipliers and are of nilpotency class $2$, a nonprobabilistic criterion for the vanishing of the Bogomolov multiplier, and establishing a sequence of Bogomolov's absolute $\gamma$-minimal factors which are $2$-groups of arbitrarily large nilpotency class, thus providing counterexamples to some of Bogomolov's claims. In relation to this, we fill a gap in the proof of triviality of Bogomolov multipliers of finite simple groups.