## 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 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 language | English |
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Pages (from-to) | 599-641 |

Number of pages | 43 |

Journal | Advances in Mathematics |

Volume | 339 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

## Keywords

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