Conjecture 8.8 — open problem in algebra

Conjecture 8.8

Does there exist a non-cyclic finitely presented group G which contains an element a such that each element of G is conjugate to some power of a? Here a power of a means a^n for some n ∈ ℤ.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement kourovka_8_8b as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `kourovka_8_8b`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 1 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-kourovka-8-8,
  title        = {Conjecture 8.8},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/kourovka-8-8}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Does there exist a non-cyclic finitely presented group $G$ which contains an element $a$ such that each element of $G$ is conjugate to some power of $a$ ?

Here a power of $a$ means $a^{n}$ for some $n \in Z$ .

by R. I. Grigorchuk

Both parts of the problem are due to D. V. Anosov.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Kourovka.«8_8».

theorem kourovka_8_8b :
    ∃ (G : Type) (_ : Group G), ¬ IsCyclic G ∧ Group.IsFinitelyPresented G ∧
      ∃ a : G, ∀ g : G, ∃ n : ℤ, IsConj g (a ^ n)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

The Kourovka Notebook

Source and licence

Imported from Formal Conjectures (the Kourovka notebook), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.