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 $\pi$: \[ | (1+\mathrm{I}_{\mathbb{F}_q})_{\mathrm{ab}} |=q^{\mathrm{k}(π)-1}|\mathrm{B}_0(π)|. \] 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.

9 pages