Units of group rings, the Bogomolov multiplier, and the fake degree conjecture

TL;DR

This paper uncovers a surprising relation connecting the abelianization of units in group rings, the Bogomolov multiplier, and conjugacy classes in finite p-groups, revealing counterexamples to a conjecture in representation theory.

Contribution

It establishes a novel formula linking the abelianization of units in group rings with the Bogomolov multiplier and conjugacy classes, providing counterexamples to the fake degree conjecture.

Findings

Derived a formula relating units in group rings to the Bogomolov multiplier.

Identified counterexamples to the fake degree conjecture for certain p-groups.

Connected algebraic structures with group-theoretic invariants in finite p-groups.

Abstract

Let $\pi$ be a finite $p$-group and $\mathbb{F}_q$ a finite field with $q=p^n$ elements. Denote by $\mathrm{I}_{\mathbb{F}_q}$ the augmentation ideal of the group ring $\mathbb{F}_q[\pi]$. We have found a surprising relation between the abelianization of $1+\mathrm{I}_{\mathbb{F}_q}$, the Bogomolov multiplier $\mathrm{B}_0(\pi)$ of $\pi$ and the number of conjugacy classes $\mathrm{k}(\pi)$ of $\pi$: \[ | (1+\mathrm{I}_{\mathbb{F}_q})_{\mathrm{ab}} |=q^{\mathrm{k}(\pi)-1}|\mathrm{B}_0(\pi)|. \] In particular, if $\pi$ is a finite $p$-group with a non-trivial Bogomolov multiplier, then $1+\mathrm{I}_{\mathbb{F}_q}$ is a counterexample to the fake degree conjecture proposed by M. Isaacs.