A counterexample to the first Zassenhaus conjecture

Leo Margolis, Florian Eisele

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


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 2 7⋅3 2⋅5⋅7 2⋅19 2 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.

Original languageEnglish
Pages (from-to)599-641
Number of pages43
JournalAdvances in Mathematics
Publication statusPublished - 1 Dec 2018


  • Group ring
  • Integral representations
  • Unit group
  • Zassenhaus conjecture


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