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

Published online by Cambridge University Press:  09 September 2016

JAVIER GARCÍA–RODRÍGUEZ ,
ANDREI JAIKIN–ZAPIRAIN  and
URBAN JEZERNIK
JAVIER GARCÍA–RODRÍGUEZ
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain. e-mails: zjaviz@gmail.com; andrei.jaikin@uam.es
ANDREI JAIKIN–ZAPIRAIN
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain. e-mails: zjaviz@gmail.com; andrei.jaikin@uam.es
URBAN JEZERNIK
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia. e-mail: urban.jezernik@gmail.com
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Abstract

Let 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 115 - 123
Copyright
Copyright © Cambridge Philosophical Society 2016 

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