False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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FALSE: every subgroup of a finitely generated free group is finitely generated

Statement

Every subgroup of a finitely generated free group is finitely generated.

Facts & Assumptions

Given: The false claim above.

[L1]

For a Schreier system, the nontrivial Schreier generators form a free basis of the subgroup (Under the stated choice boundary, every subgroup of a free group is free with its nontrivial Schreier generators as a basis).

Refutation

technique · direct

1.1

In the free group $F (a, b)$ , let $ϕ : F (a, b) \to Z$ send $a \mapsto 1$ and $b \mapsto 0$ , and let $H = \ker ϕ$ . The right cosets of $H$ are $H a^{n}$ for $n \in Z$ , and $T = {a^{n} : n \in Z}$ is a Schreier system.

givenconstruct

2.1

For this system, the nontrivial Schreier generators are exactly the conjugates $a^{n} b a^{- n}$ for $n \in Z$ , because $s (a^{n}, a) = 1$ and $s (a^{n}, b) = a^{n} b a^{- n}$ . By [L1], this infinite family is a free basis of $H$ .

L1step 1.1algebra

3.1

A free basis cannot be finite when it contains infinitely many distinct elements, so $H$ is not finitely generated. This subgroup of the rank-two free group $F (a, b)$ refutes the statement.

step 2.1∎

Depends on

Under the stated choice boundary, every subgroup of a free group is free with its nontrivial Schreier generators as a basis

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)