Class-preserving automorphisms

DefinitionClaim of Kourovka Problem 16.62#✓fully proved

For every group \(G\): if every automorphism of \(G\) sends each element \(g \in G\) to a conjugate of \(g\) (equivalently, maps every conjugacy class of \(G\) into itself), then every automorphism of \(G\) is inner, that is, of the form \(x \mapsto h x h^{-1}\) for some \(h \in G\).

def classPreservingAutomorphismsInner : Prop :=
  ∀ (G : Type) [Group G], (∀ (α : MulAut G) (g : G), IsConj (α g) g) →
    ∀ α : MulAut G, ∃ h : G, α = MulAut.conj h

Used by 1

Used byGroups whose automorphisms all preserve conjugacy classes can have outer automorphisms

TheoremGroups whose automorphisms all preserve conjugacy classes can have outer automorphisms#●ready to prove

The claim of Kourovka Problem 16.62 is false: there is a group \(G\) such that every automorphism of \(G\) sends each element \(g \in G\) to a conjugate of \(g\), but some automorphism of \(G\) is not inner, that is, not of the form \(x \mapsto h x h^{-1}\) for any \(h \in G\).

Heineken constructed finite \(p\)-groups all of whose automorphisms preserve every conjugacy class, and Gaschütz' theorem gives these groups outer automorphisms.

public theorem_wanted classPreservingAutomorphismsInner_refuted :
    ¬ classPreservingAutomorphismsInner

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 16.62 (D. MacHale); resolution: H. Heineken, Arch. Math. (Basel) 33 (1979/80), no. 6, 497-503; A. Mann, letter of 20 April 2006, as recorded in the Kourovka Archive, Kourovka Archive, Problem 16.62 (Not necessarily: Heineken's finite p-groups all of whose automorphisms preserve all conjugacy classes have outer automorphisms by Gaschütz' theorem).

Uses 1

Statement usesClaim of Kourovka Problem 16.62

DefinitionClaim of Scottish Book Problem 112#✓fully proved

The claim that for every group \(G\), every automorphism \(\alpha\) of \(G\) that sends each element to a conjugate of itself (for each \(x \in G\) there is \(g \in G\) with \(\alpha(x) = g x g^{-1}\)) is inner, that is, there is a single \(g \in G\) with \(\alpha(x) = g x g^{-1}\) for all \(x \in G\).

def ScottishBookProblem112Claim : Prop :=
  ∀ (G : Type*) [Group G] (α : MulAut G),
    (∀ x : G, ∃ g : G, α x = g * x * g⁻¹) →
    ∃ g : G, ∀ x : G, α x = g * x * g⁻¹

Used by 1

Used byClass-preserving automorphisms need not be inner

TheoremClass-preserving automorphisms need not be inner#●ready to prove

The claim of Scottish Book Problem 112 is false: there are a group \(G\) and an automorphism \(\alpha\) of \(G\) that sends every element to a conjugate of itself (for each \(x \in G\) there is \(g \in G\) with \(\alpha(x) = g x g^{-1}\)) but is not inner (there is no \(g \in G\) with \(\alpha(x) = g x g^{-1}\) for all \(x \in G\)).

Burnside's 1913 example is such a group: for an odd prime \(p\), a group of order \(p^6\), isomorphic to the group of \(3 \times 3\) lower unitriangular matrices over the field \(\mathbb{F}_{p^2}\), whose group of class-preserving outer automorphisms is nontrivial.

theorem_wanted scottishBook112_refuted : ¬ ScottishBookProblem112Claim

Sources

• W. Burnside's 1913 example, as summarized in M. K. Yadav, "Class preserving automorphisms of finite p-groups: A survey", arXiv:1002.1359, §1 (Burnside constructed a group of order p ^ 6 with Out_c (G) ≠ 1).

Uses 1

Statement usesClaim of Scottish Book Problem 112

2 fully proved2 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.