Groups with few maximal subgroups

DefinitionBergman's question for three maximal subgroups#✓fully proved

Kourovka 17.17 (G. M. Bergman) asks whether a finitely generated group with only finitely many maximal subgroups must be finite. This is the case of three maximal subgroups, which the problem singles out: every finitely generated group with exactly three maximal subgroups is finite.

def threeMaximalSubgroupsFinite : Prop :=
  ∀ (G : Type) [Group G], Group.FG G → {M : Subgroup G | IsCoatom M}.ncard = 3 → Finite G

Used by 1

Used byNot every finitely generated group with three maximal subgroups is finite

TheoremNot every finitely generated group with three maximal subgroups is finite#●ready to prove

There is a finitely generated infinite group with exactly three maximal subgroups; that is, it is false that every finitely generated group with exactly three maximal subgroups is finite.

This answers Kourovka Problem 17.17 (G. M. Bergman) in the negative: the Kourovka Archive cites a 2-generated group constructed in Section 7 of Ershov and Jaikin-Zapirain (2013), which is infinite and has exactly three maximal subgroups, each of index 2.

public theorem_wanted threeMaximalSubgroupsFinite_refuted : ¬ threeMaximalSubgroupsFinite

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 17.17 (G. M. Bergman); resolution: M. Ershov and A. Jaikin-Zapirain, J. Reine Angew. Math. 677 (2013), 71-134, Kourovka Archive, Problem 17.17 (No, it need not: a 2-generated 2-LERF group in Section 7 of Ershov-Jaikin-Zapirain is an infinite group with 3 maximal subgroups, all of index 2).

Uses 1

Statement usesBergman's question for three maximal subgroups

1 fully proved1 ready to prove

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