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The Zassenhaus conjecture for torsion units in integral group rings

From papers

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

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

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

Progress summary

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Equivalent formulations 1

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

  1. Zassenhaus conjecture for torsion units in integral group rings

    Let GG be a finite group and let uu be a torsion unit in the group of normalized units V(ZG)\mathrm{V}(\mathbb{Z}G), namely the units of augmentation one in the integral group ring ZG\mathbb{Z}G. Zassenhaus conjecture. There exists a unit xx in QG\mathbb{Q}G such that

    x−1ux=gx^{-1}ux=g

    for some g∈Gg\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).

Sources & referencesView supporting material

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.

finite-groupsgroup-ringsintegral-group-ringstorsion-unitsunitary-representations

Solutions 0

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