Groups isomorphic to their normal subgroups

DefinitionHall's question on groups isomorphic to their nontrivial normal subgroups (Kourovka 12.45)#✓fully proved

The statement that every nontrivial group that is isomorphic to each of its nontrivial normal subgroups is free of infinite rank (isomorphic to the free group on an infinite set), simple, or infinite cyclic.

def isoToNormalSubgroupsFreeSimpleOrCyclic : Prop :=
  ∀ (G : Type) [Group G], Nontrivial G →
    (∀ N : Subgroup G, N.Normal → N ≠ ⊥ → Nonempty (N ≃* G)) →
    (∃ X : Type, Infinite X ∧ Nonempty (FreeGroup X ≃* G)) ∨ IsSimpleGroup G ∨
      (IsCyclic G ∧ Infinite G)

Used by 1

Used byA group isomorphic to its nontrivial normal subgroups need not be free of infinite rank, simple or infinite cyclic

TheoremA group isomorphic to its nontrivial normal subgroups need not be free of infinite rank, simple or infinite cyclic#●ready to prove

A group isomorphic to its nontrivial normal subgroups need not be free of infinite rank, simple or infinite cyclic. There is a nontrivial group \(G\) that is isomorphic to each of its nontrivial normal subgroups and is neither free of infinite rank (isomorphic to the free group on an infinite set), nor simple, nor infinite cyclic. That is, the answer to Kourovka Problem 12.45 is negative (V. N. Obraztsov, 1997).

public theorem_wanted isoToNormalSubgroupsFreeSimpleOrCyclic_refuted :
    ¬ isoToNormalSubgroupsFreeSimpleOrCyclic

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 12.45 (P. Hall; J. C. Lennox); resolution: V. N. Obraztsov, Proc. London Math. Soc. (3) 75 (1997), 79-98, Kourovka Archive, Problem 12.45: No.

Uses 1

Statement usesHall's question on groups isomorphic to their nontrivial normal subgroups (Kourovka 12.45)

1 fully proved1 ready to prove

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