Retracts of free groups

DefinitionRetract of a group#✓fully proved

A subgroup \(R\) of a group \(F\) is a retract of \(F\) when there is an endomorphism \(r \colon F \to F\) whose image is \(R\) and which fixes every element of \(R\).

def IsRetract {F : Type*} [Group F] (R : Subgroup F) : Prop :=
  ∃ r : F →* F, r.range = R ∧ ∀ x ∈ R, r x = x

Used by 1

Used byBergman's retract intersection question (Kourovka Problem 17.19)

DefinitionBergman's retract intersection question (Kourovka Problem 17.19)#✓fully proved

The statement that for every natural number \(n\), every retract \(R\) of the free group \(F_n\) on \(n\) generators and every finitely generated subgroup \(H\) of \(F_n\), the intersection \(H \cap R\) is a retract of \(H\).

A subgroup of a free group is finitely generated exactly when it has finite rank.

def retractIntersectionIsRetract : Prop :=
  ∀ (n : ℕ) (R H : Subgroup (FreeGroup (Fin n))), IsRetract R → H.FG →
    IsRetract ((H ⊓ R).subgroupOf H)

Uses 1 · Used by 1

Statement usesRetract of a group

Used byNegative answer to Bergman's retract intersection question (Kourovka Problem 17.19)

TheoremNegative answer to Bergman's retract intersection question (Kourovka Problem 17.19)#●ready to prove

It is not the case that for every natural number \(n\), every retract \(R\) of the free group \(F_n\) on \(n\) generators and every finitely generated subgroup \(H\) of \(F_n\), the intersection \(H \cap R\) is a retract of \(H\).

Snopce, Tanushevski and Zalesskii (2022) give a free group \(F\) of finite rank, a retract \(R\) of \(F\) and a finitely generated subgroup \(H\) of \(F\) such that \(H \cap R\) is not a retract of \(H\); the Kourovka Archive answers 17.19 "No, it need not".

public theorem_wanted retractIntersectionIsRetract_refuted : ¬ retractIntersectionIsRetract

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 17.19 (G. M. Bergman); resolution: I. Snopce, S. Tanushevski and P. Zalesskii, Int. Math. Res. Notices 2022, no. 11 (2022), 8280-8294, Kourovka Archive, Problem 17.19 (No, it need not).

Uses 1

Statement usesBergman's retract intersection question (Kourovka Problem 17.19)

2 fully proved1 ready to prove

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