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

Abstract

AbstractLet π be a finite p-group and ${\mathbb{F}_{q}}$ a finite field with q = pn elements. Denote by $\I_{\mathbb{F}_{q}}$ the augmentation ideal of the group ring ${\mathbb{F}_{q}}$[π]. We have found a surprising relation between the abelianization of 1 + $\I_{\mathbb{F}_{q}}$, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π: $$ \left | (1+\I_{\Fq})_{\ab} \right |=q^{\kk(\pi)-1}|\!\B_0(\pi)|. In particular, if π is a finite p-group with a non-trivial Bogomolov multiplier, then 1 + $\I_{\mathbb{F}_{q}}$ is a counterexample to the fake degree conjecture proposed by M. Isaacs.