A Counterexample to the First Zassenhaus Conjecture

TL;DR

This paper disproves the first Zassenhaus conjecture by constructing an infinite family of counterexamples, including a specific metabelian group, showing units of finite order in the integral group ring are not always conjugate to group elements in the rational group algebra.

Contribution

It provides the first known counterexamples to the Zassenhaus conjecture, including an explicit infinite family and a minimal example involving a complex metabelian group.

Findings

Counterexamples to the Zassenhaus conjecture are constructed.

Units of certain orders in integral group rings are not conjugate to group elements.

An explicit minimal example involving a large metabelian group is provided.

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 $\pm g$.