The status of the Zassenhaus conjecture for small groups

TL;DR

This paper investigates the Zassenhaus conjecture for small groups up to order 288, using computational and theoretical methods to verify the conjecture for most cases and identifying unresolved instances.

Contribution

It introduces two new computational methods, the quotient method and the partially central unit construction, to verify the Zassenhaus conjecture for small groups.

Findings

Verified the conjecture for all groups of order less than 144.

Identified unresolved cases among groups of orders 144 to 287.

Developed new computational techniques for group ring analysis.

Abstract

We identify all small groups of order up to 288 in the GAP Library for which the Zassenhaus conjecture on rational conjugacy of units of finite order in the integral group ring cannot be established by an existing method. The groups must first survive all theoretical sieves and all known restrictions on partial augmentations (the HeLP$^+$ method). Then two new computational methods for verifying the Zassenhaus conjecture are applied to the unresolved cases, which we call the quotient method and the partially central unit construction method. To the cases that remain we attempt an assortment of special arguments available for units of certain orders and the lattice method. In the end, the Zassenhaus conjecture is verified for all groups of order less than 144 and we give a list of all remaining cases among groups of orders 144 to 287.