A Counterexample to the First Zassenhaus Conjecture

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 $a^{-1}\cdot u \cdot a=\pm g$ 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 $\pm g$.