Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Algebra P-kourovka-8-8

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.

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.

Start working on it Submit a claim Follow
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 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

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

Here a power of means for some .

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». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem kourovka_8_8b : answer(sorry) ↔
    ∃ (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 e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.