Zassenhaus conjectures

DefinitionZassenhaus conjecture on torsion units#✓fully proved

Kourovka 17.67 (H. Zassenhaus's conjecture, recorded by V. D. Mazurov): every unit of finite order of the integral group ring ℤG of a finite group G is conjugate, by a unit of the rational group ring ℚG, to an element ±g with g ∈ G.

def zassenhausConjecture : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (u : (MonoidAlgebra ℤ G)ˣ), IsOfFinOrder u →
    ∃ (v : (MonoidAlgebra ℚ G)ˣ) (g : G) (ε : ℚ), (ε = 1 ∨ ε = -1) ∧
      (v : MonoidAlgebra ℚ G) *
          MonoidAlgebra.mapRingHom G (Int.castRingHom ℚ) (u : MonoidAlgebra ℤ G) *
          (↑v⁻¹ : MonoidAlgebra ℚ G) = MonoidAlgebra.single g ε

Used by 1

Used byCounterexample to the first Zassenhaus conjecture (Kourovka 17.67)

TheoremCounterexample to the first Zassenhaus conjecture (Kourovka 17.67)#●ready to prove

There exist a finite group \(G\) and a unit \(u\) of finite order of the integral group ring \(\mathbb{Z}G\) such that for every unit \(v\) of the rational group ring \(\mathbb{Q}G\), every \(g \in G\) and every sign \(\varepsilon \in \{1, -1\}\), the conjugate \(v u v^{-1}\), computed in \(\mathbb{Q}G\), is not equal to \(\varepsilon g\). That is, the first Zassenhaus conjecture is false.

Eisele and Margolis (abstract and Theorem A) give a metabelian group \(G\) of order \(2^7 \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 19^2\) and a unit of order \(7 \cdot 19\) in \(\mathbb{Z}G\) that is not conjugate in \(\mathbb{Q}G\) to any element \(\pm g\), \(g \in G\); the Kourovka Archive records for 17.67 "A counterexample was constructed".

public theorem_wanted zassenhausConjecture_refuted : ¬ zassenhausConjecture

def zassenhausConjecture : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (u : (MonoidAlgebra ℤ G)ˣ), IsOfFinOrder u →
    ∃ (v : (MonoidAlgebra ℚ G)ˣ) (g : G) (ε : ℚ), (ε = 1 ∨ ε = -1) ∧
      (v : MonoidAlgebra ℚ G) *
          MonoidAlgebra.mapRingHom G (Int.castRingHom ℚ) (u : MonoidAlgebra ℤ G) *
          (↑v⁻¹ : MonoidAlgebra ℚ G) = MonoidAlgebra.single g ε

WantedExt/GroupTheory/ZassenhausConjectureKourovka17Problem67Wanted.lean:40

Sources

E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 17.67 (V. D. Mazurov); resolution: F. Eisele and L. Margolis, A counterexample to the first Zassenhaus conjecture, Adv. Math. 339 (2018), 599-641, abstract and Theorem A (a metabelian group of order 2⁷·3²·5·7²·19² whose integral group ring contains a unit of order 7·19 not conjugate in the rational group algebra to any ±g); Kourovka Archive, Problem 17.67 (A counterexample was constructed), http://eprints.gla.ac.uk/177223/7/177223.pdf.

Uses 1

Statement usesZassenhaus conjecture on torsion units

1 fully proved1 ready to prove

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