Let G be a finite group, let ZG be its integral group ring, and let U( ZG) denote its group of units. A unit has finite order if it has finite multiplicative order. Zassenhaus conjecture. If u∈U( ZG) is a unit of finite order, then u is conjugate within QG to pm g for some g∈ G. This conjecture concerns the structure of finite-order units in integral group rings. The paper proves it when the Sylow p-subgroup of G has order p, and gives further positive results for certain groups operatornamePSL(2,q); however, it is false in a local form, and its general validity remains open.

The Zassenhaus conjecture for torsion units in integral group rings

About 11 years old · traced to 2015

Let $G$ be a finite group, let $\mathbb ZG$ be its integral group ring, and let $\mathcal{U}(\mathbb ZG)$ denote its group of units. A unit has finite order if it has finite multiplicative order.

Zassenhaus conjecture. If $u\in\mathcal{U}(\mathbb ZG)$ is a unit of finite order, then $u$ is conjugate within $\mathbb QG$ to $\pm g$ for some $g\in G$ .

This conjecture concerns the structure of finite-order units in integral group rings. The paper proves it when the Sylow $p$ -subgroup of $G$ has order $p$ , and gives further positive results for certain groups $\operatorname{PSL}(2,q)$ ; however, it is false in a local form, and its general validity remains open.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Zassenhaus conjecture for torsion units in integral group rings
Let $G$ be a finite group and let $u$ be a torsion unit in the group of normalized units $\mathrm{V}(\mathbb{Z}G)$ , namely the units of augmentation one in the integral group ring $\mathbb{Z}G$ . Zassenhaus conjecture. There exists a unit $x$ in $\mathbb{Q}G$ such that
$x^{-1}ux=g$
for some $g\in G$ . The conjecture is one of the main open questions concerning torsion units in integral group rings; it asserts that every normalized torsion unit is rationally conjugate to a group element.
source: Andreas Bächle and Leo Margolis, “HeLP – A GAP-package for torsion units in integral group rings”, arXiv:1507.08174 (2016).

References

Primary source

F. Eisele and L. Margolis, “Units in Blocks of Defect 1 and the Zassenhaus Conjecture”, arXiv:2212.06634 (2022).

Additional references

2 papers in this index state this conjecture (2015–2022). The statement above is taken from the most recent of them; the others are arXiv:1507.08174.

Posted 1 month ago

The Zassenhaus conjecture for torsion units in integral group rings

Zassenhaus conjecture for torsion units in integral group rings

Primary source

Additional references

Progress summary

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