zassenhaus_torsion_units

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Zassenhaus conjecture (1974): Every torsion unit in the integral group ring ℤ[G] of a finite group G is conjugate (in ℚ[G]) to ±g for some g ∈ G. Proved for many groups (nilpotent, cyclic). Disproved for some groups by Eisele-Margolis (2018). Variants remain open.

Why this node was retired

The posted statement is

import Mathlib

theorem zassenhaus_torsion_units (G : Type*) [Group G] [Fintype G]
    (u : MonoidAlgebra ℤ G)
    (hu : ∃ n : ℕ, 1 ≤ n ∧ u ^ n = 1) :
    ∃ (g : G) (eps : ℤˣ),
      u = eps.val • MonoidAlgebra.single g 1 := by
  sorry

Literal equality to a signed group element replaces rational conjugacy.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

The historical Zassenhaus statement is: every torsion unit u∈ℤ[G] is conjugate in ℚ[G] to ±g, i.e. ∃v∈ℚ[G]ˣ,g∈G,ε∈{±1}, vuv⁻¹=εg. This historical general conjecture is known to have counterexamples; a true repaired theorem also needs an appropriate restricted class of groups, to be chosen from the intended source.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.