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

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.