TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every finitely generated free group is subgroup separable

Statement

Every finitely generated free group is subgroup separable.

Facts & Assumptions

Given: A finitely generated free group $F$ , a finitely generated subgroup $H \leq F$ , and an element $g \in F ∖ H$ .

[F1]

Marshall Hall's theorem identifies finitely generated subgroups of finitely generated free groups as closed in the profinite topology; equivalently, they are subgroup separable.

Proof

technique · direct

1.1

The present hypotheses are exactly the subgroup-separability conclusion of [F1]. Therefore there exists a finite-index subgroup of $F$ that contains $H$ but not $g$ . Since $g \in F ∖ H$ was arbitrary, every finitely generated subgroup of $F$ is closed in the profinite topology.

F1given

2.1

This is precisely the definition of subgroup separability from A subgroup is separable when it is closed in the profinite topology, and a group is LERF when every finitely generated subgroup is separable. Hence every finitely generated free group is subgroup separable.

step 1.1∎

Depends on

A subgroup is separable when it is closed in the profinite topology, and a group is LERF when every finitely generated subgroup is separable

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)
Rita Gitik and Eliyahu Rips, On separability properties of free groups (standard reference, not scraped)