ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Marshall Hall's theorem produces a separating finite-index overgroup

Example

In $F (a, b)$ , let $H = ⟨ b, a^{2} ⟩$ and $w = a$ . Then the index-two subgroup

$K = ⟨ b, a^{2}, a b a^{- 1} ⟩$

contains $H$ as a free factor and omits $w$ .

Facts & Assumptions

Given: The subgroup $H = ⟨ b, a^{2} ⟩$ of $F (a, b)$ and the element $w = a$ .

[L1]

Every finitely generated subgroup of a finite-rank free group is a free factor of some finite-index subgroup (Every finitely generated subgroup of a finite-rank free group is a free factor of a finite-index subgroup).

Verification

technique · direct

1.1givenconstruct

The subgroup $K$ consists of words with even exponent sum in $a$ , so it has index $2$ in $F (a, b)$ . The basis computation $K = ⟨ b, a^{2}, a b a^{- 1} ⟩$ shows that $H$ is generated by two members of that free basis.

2.1L1step 1.1∎

Hence $K$ splits as the free product of $H$ with the cyclic subgroup generated by $a b a^{- 1}$ , so $H$ is a free factor of $K$ . Also $a \notin K$ because its exponent sum in $a$ is odd. This is a concrete separating finite-index overgroup of the kind promised abstractly by [L1].

Depends on

Every finitely generated subgroup of a finite-rank free group is a free factor of a finite-index subgroup

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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