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A counterexample to the first Zassenhaus conjecture

Eisele, F. ORCID: 0000-0001-8267-2094 and Margolis, L. (2018). A counterexample to the first Zassenhaus conjecture. Adavances in Mathematics, 339, pp. 599-641. doi: 10.1016/j.aim.2018.10.004

Abstract

Hans J. Zassenhaus conjectured that for any unit u of finite order in the integral group ring of a finite group G there exists a unit a in the rational group algebra of G such that a−1· u · a = ±g for some g ∈ G. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order 27·32·5·72·192 whose integral group ring contains a unit of order 7 · 19 which, in the rational group algebra, is not conjugate to any element of the form ±g.

Publication Type: Article
Additional Information: © Elsevier 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords: Unit group, Group ring, Zassenhaus conjecture, Integral representations
Subjects: Q Science > QA Mathematics
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
Date Deposited: 13 Jan 2020 14:24
URI: https://openaccess.city.ac.uk/id/eprint/23439
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