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 $a^{-1}\cdot u\cdot a=\pm g$ for some $g\in G$. 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 $2^7\cdot 3^2\cdot 5\cdot 7^2\cdot {19}^2$ whose integral group ring contains a unit of order $7\cdot 19$ which, in the rational group algebra, is not conjugate to any element of the form $\pm g$.