An algorithm to construct candidates to counterexamples to the Zassenhaus Conjecture

TL;DR

This paper develops algorithms based on Cliff-Weiss inequalities to identify potential counterexamples to the Zassenhaus Conjecture, providing the first known counterexamples and advancing understanding of units in integral group rings.

Contribution

The authors introduce new algorithms to analyze Sehgal's Problem for groups with at most one non-abelian Sylow subgroup, leading to the construction of explicit counterexamples to the Zassenhaus Conjecture.

Findings

Algorithms successfully identify candidates for counterexamples.

Construction of explicit metabelian groups as potential counterexamples.

Confirmed some candidates as actual counterexamples to the conjecture.

Abstract

Let $G$ be a finite group, $N$ a nilpotent normal subgroup of $G$ and let $\mathrm{V}(\mathbb{\Z} G, N)$ denote the group formed by the units of the integral group ring $\mathbb{\Z} G$ of $G$ which map to the identity under the natural homomorphism $\mathbb{\Z} G \rightarrow \mathbb{\Z} (G/N)$. Sehgal asked whether any torsion element of $\mathrm{V}(\mathbb{\Z} G, N)$ is conjugate in the rational group algebra of $G$ to an element of $G$. This is a special case of the Zassenhaus Conjecture. By results of Cliff and Weiss and Hertweck, Sehgal's Problem has a positive solution if $N$ has at most one non-cyclic Sylow subgroup. We present some algorithms to study Sehgal's Problem when $N$ has at most one non-abelian Sylow subgroup. They are based on the Cliff-Weiss inequalities introduced by the authors in a previous paper. With the help of these algorithms we obtain some positive answers to Sehgal's Problem and use them to show that for units in $\mathrm{V}(\mathbb{\Z} G,N)$ our method is strictly stronger than the well known HeLP Method. We then present a method to use the output of one of the algorithms to construct explicit metabelian groups which are candidates to a negative solution to Sehgal's Problem. Recently Eisele and Margolis showed that some of the examples proposed in this paper are indeed counterexamples to the Zassenhaus Conjecture. These are the first known counterexamples. Moreover, we prove that every metabelian negative solution of Sehgal's Problem satisfying some minimal conditions is given by our construction.