Twisted conjugacy classes

DefinitionTwisted conjugacy class of the identity (Kourovka 18.15)#✓fully proved

For a group \(G\) and an automorphism \(\varphi\) of \(G\), the \(\varphi\)-twisted conjugacy class of the identity is the set \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\} \subseteq G\).

def twistedUnitClass {G : Type} [Group G] (φ : G ≃* G) : Set G :=
  Set.range fun g : G => g⁻¹ * φ g
Used by 1
Used byKourovka Problem 18.15: an inner twisted class that is not a subgroup

DefinitionKourovka Problem 18.15: an inner twisted class that is not a subgroup#✓fully proved

Every nontrivial group \(G\) with trivial centre has an element \(a \in G\) such that, for the inner automorphism \(\varphi(g) = aga^{-1}\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is not a subgroup of \(G\).

The conjecture is due to V. G. Bardakov, M. V. Neshchadim and T. R. Nasybullov.

def trivialCentreHasInnerTwistedClassNotSubgroup : Prop :=
  ∀ (G : Type) [Group G], Nontrivial G → Subgroup.center G = ⊥ →
    ∃ a : G, ¬ ∃ H : Subgroup G, (H : Set G) = twistedUnitClass (MulAut.conj a)
Uses 1 · Used by 1
Statement usesTwisted conjugacy class of the identity (Kourovka 18.15)
Used byNegative answer to Kourovka Problem 18.15

TheoremNegative answer to Kourovka Problem 18.15#●ready to prove

The conjecture of Kourovka Problem 18.15 is false: there is a nontrivial group \(G\) with trivial centre such that for every \(a \in G\), with \(\varphi\) the inner automorphism \(\varphi(g) = aga^{-1}\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is a subgroup of \(G\).

The Kourovka Archive answers 18.15 "No, not always", citing V. V. Bludov's Example 2 in Gonçalves–Nasybullov (2019), a group with trivial centre in which \([e]_\varphi\) is a subgroup for every inner automorphism \(\varphi\).

public theorem_wanted trivialCentreHasInnerTwistedClassNotSubgroup_refuted :
    ¬ trivialCentreHasInnerTwistedClassNotSubgroup
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 18.15 (V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov); resolution: V. V. Bludov, Example 2 in D. Gonçalves and T. Nasybullov, On groups where the twisted conjugacy class of the unit element is a subgroup, Comm. Algebra 47 (2019), no. 3, 930-944, Kourovka Archive, Problem 18.15: No, not always; see Bludov's Example 2, https://arxiv.org/abs/1401.0300v49.
Uses 1
Statement usesKourovka Problem 18.15: an inner twisted class that is not a subgroup

DefinitionTwisted conjugacy class of the identity (Kourovka 18.14)#✓fully proved

For a group \(G\) and an automorphism \(\varphi\) of \(G\), the \(\varphi\)-twisted conjugacy class of the identity is the set \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\} \subseteq G\).

def twistedUnitClass {G : Type} [Group G] (φ : G ≃* G) : Set G :=
  Set.range fun g : G => g⁻¹ * φ g
Used by 1
Used byGroups whose twisted classes of the identity are all subgroups

DefinitionGroups whose twisted classes of the identity are all subgroups#✓fully proved

A group \(G\) has this property when, for every automorphism \(\varphi\) of \(G\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is a subgroup of \(G\).

def AllTwistedUnitClassesSubgroups (G : Type) [Group G] : Prop :=
  ∀ φ : G ≃* G, ∃ H : Subgroup G, (H : Set G) = twistedUnitClass φ
Uses 1 · Used by 2
Statement usesTwisted conjugacy class of the identity (Kourovka 18.14)
Used byKourovka Problem 18.14, abelian assertion for finitely generated groups · Kourovka Problem 18.14, nilpotency assertion

DefinitionKourovka Problem 18.14, nilpotency assertion#✓fully proved

Every group \(G\) in which, for every automorphism \(\varphi\) of \(G\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is a subgroup of \(G\), is nilpotent.

This is the first assertion of the conjecture of V. G. Bardakov, M. V. Neshchadim and T. R. Nasybullov.

def twistedUnitClassesSubgroupsImpliesNilpotent : Prop :=
  ∀ (G : Type) [Group G], AllTwistedUnitClassesSubgroups G → Group.IsNilpotent G
Uses 1 · Used by 1
Statement usesGroups whose twisted classes of the identity are all subgroups
Used byNegative answer to the nilpotency assertion of Kourovka Problem 18.14

DefinitionKourovka Problem 18.14, abelian assertion for finitely generated groups#✓fully proved

Every finitely generated group \(G\) in which, for every automorphism \(\varphi\) of \(G\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is a subgroup of \(G\), is abelian.

This is the second assertion of the conjecture of Kourovka Problem 18.14.

def twistedUnitClassesSubgroupsFGImpliesAbelian : Prop :=
  ∀ (G : Type) [Group G], Group.FG G → AllTwistedUnitClassesSubgroups G →
    ∀ x y : G, x * y = y * x
Uses 1 · Used by 1
Statement usesGroups whose twisted classes of the identity are all subgroups
Used byNegative answer to the abelian assertion of Kourovka Problem 18.14

TheoremNegative answer to the nilpotency assertion of Kourovka Problem 18.14#●ready to prove

There is a group \(G\) that is not nilpotent although, for every automorphism \(\varphi\) of \(G\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is a subgroup of \(G\).

Nicotera's Corollary 3.1 gives such an infinite group, a restricted direct product of the finite groups of Theorem 1.2 for pairwise distinct primes.

public theorem_wanted twistedUnitClassesSubgroupsImpliesNilpotent_refuted :
    ¬ twistedUnitClassesSubgroupsImpliesNilpotent
Sources
  • E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.14 (V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov); resolution: C. Nicotera, On finite groups in which the twisted conjugacy classes of the unit element are subgroups, Arch. Math. 123 (2024), no. 3, 225-232, Theorem 1.2 (for every n and every odd prime p, a finite p-group of class n in which [G, φ] is a subgroup for every φ ∈ Aut G) and Corollary 3.1 (an infinite non-nilpotent group with this property); the Kourovka Notebook (21st issue) marks 18.14 solved, https://doi.org/10.1007/s00013-024-02025-6.
Uses 1
Statement usesKourovka Problem 18.14, nilpotency assertion

TheoremNegative answer to the abelian assertion of Kourovka Problem 18.14#●ready to prove

There is a finitely generated group \(G\) that is not abelian although, for every automorphism \(\varphi\) of \(G\), the twisted conjugacy class of the identity \([e]_\varphi = \{g^{-1}\varphi(g) : g \in G\}\) is a subgroup of \(G\).

Nicotera's Theorem 1.2 gives, for every \(n\) and every odd prime \(p\), a finite \(p\)-group of nilpotency class \(n\) with this property; for \(n \ge 2\) it is not abelian.

public theorem_wanted twistedUnitClassesSubgroupsFGImpliesAbelian_refuted :
    ¬ twistedUnitClassesSubgroupsFGImpliesAbelian
Sources
  • E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.14 (V. G. Bardakov, M. V. Neshchadim, T. R. Nasybullov); resolution: C. Nicotera, On finite groups in which the twisted conjugacy classes of the unit element are subgroups, Arch. Math. 123 (2024), no. 3, 225-232, Theorem 1.2 (for every n and every odd prime p, a finite p-group of class n in which [G, φ] is a subgroup for every φ ∈ Aut G) and Corollary 3.1 (an infinite non-nilpotent group with this property); the Kourovka Notebook (21st issue) marks 18.14 solved, https://doi.org/10.1007/s00013-024-02025-6.
Uses 1
Statement usesKourovka Problem 18.14, abelian assertion for finitely generated groups
6 fully proved3 ready to prove

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