Outer automorphisms of poly-infinite-cyclic groups

DefinitionNormal series with infinite cyclic factors#✓fully proved

A group \(G\) has a normal series with infinite cyclic factors when there are a natural number \(n\) and subgroups \(1 = H_0 \le H_1 \le \cdots \le H_n = G\), each normal in \(G\), such that every factor \(H_{i+1}/H_i\) is infinite cyclic.

def HasNormalSeriesInfiniteCyclicFactors (G : Type*) [Group G] : Prop :=
  ∃ (n : ℕ) (H : Fin (n + 1) → Subgroup G), H 0 = ⊥ ∧ H (Fin.last n) = ⊤ ∧
    (∀ i, (H i).Normal) ∧ ∀ i : Fin n, H i.castSucc ≤ H i.succ ∧
      ∃ _ : ((H i.castSucc).subgroupOf (H i.succ)).Normal,
        IsCyclic (H i.succ ⧸ (H i.castSucc).subgroupOf (H i.succ)) ∧
          Infinite (H i.succ ⧸ (H i.castSucc).subgroupOf (H i.succ))

Used by 1

Used byOuter automorphisms of groups with an infinite cyclic normal series (Kourovka 13.23)

DefinitionOuter automorphisms of groups with an infinite cyclic normal series (Kourovka 13.23)#✓fully proved

The conjecture that every nontrivial group with a normal series with infinite cyclic factors has an automorphism that is not inner.

This is Kourovka Problem 13.23 (F. de Giovanni).

def polyInfiniteCyclicHasOuterAutomorphism : Prop :=
  ∀ (G : Type) [Group G], Nontrivial G → HasNormalSeriesInfiniteCyclicFactors G →
    ∃ φ : MulAut G, ∀ g : G, φ ≠ MulAut.conj g

Uses 1 · Used by 1

Statement usesNormal series with infinite cyclic factors

Used byOnly inner automorphisms despite an infinite cyclic normal series (Kourovka 13.23 refuted)

TheoremOnly inner automorphisms despite an infinite cyclic normal series (Kourovka 13.23 refuted)#●ready to prove

There is a nontrivial group \(G\) with a normal series with infinite cyclic factors such that every automorphism of \(G\) is inner. So the conjecture of Kourovka Problem 13.23 is false.

Brescia, Ingrosso and Trombetti construct a torsion-free group \(G\) of Hirsch length \(14\) with a series \(1 = G_0 \trianglelefteq \cdots \trianglelefteq G_{14} = G\) of normal subgroups of \(G\) with infinite cyclic factors and \(\mathrm{Out}(G) = 1\); the Kourovka Notebook (21st issue) marks 13.23 solved, "No, not always".

public theorem_wanted polyInfiniteCyclicHasOuterAutomorphism_refuted :
    ¬ polyInfiniteCyclicHasOuterAutomorphism

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 13.23 (F. de Giovanni); resolution: M. Brescia, E. Ingrosso and M. Trombetti, A torsion-free supersoluble group with trivial outer automorphism group, preprint (2026), arXiv:2607.11775, abstract: a torsion-free group of Hirsch length 14 with a series 1 = G_0 ◁ ⋯ ◁ G_14 = G of normal subgroups of G with infinite cyclic factors and Out(G) = 1; the Kourovka Notebook (21st issue) marks 13.23 solved, https://arxiv.org/abs/2607.11775.

Uses 1

Statement usesOuter automorphisms of groups with an infinite cyclic normal series (Kourovka 13.23)

2 fully proved1 ready to prove

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