A counterexample to the first Zassenhaus conjecture

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