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Level A · Machine-checkable Hard Algebra P-leinster-group

Leinster Groups

Conjecture: Are there infinitely many Leinster groups? This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups. Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".

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-leinster-group,
  title        = {Leinster Groups},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/leinster-group}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Refuted
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Current state

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

The problem

The question

Conjecture: Are there infinitely many Leinster groups?

This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups.

Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".

A finite group is a Leinster group if the sum of the orders of all its normal subgroups equals twice the group's order.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.LeinsterGroup. answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem infinitely_many_leinster_groups : answer(sorry) ↔
    ¬∃ n : ℕ, ∀ G : Type, ∀ (_ : Group G) (_ : Fintype G),
      IsLeinster G → Fintype.card G < 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

TODO: The following properties from the Wikipedia article can also be formalized:

  • There are no Leinster groups that are symmetric or alternating.
  • There is no Leinster group of order p²q² where p, q are primes.
  • No finite semi-simple group is Leinster.
  • No p-group can be a Leinster group.
  • All abelian Leinster groups are cyclic with order equal to a perfect number.

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.