False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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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.

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FALSE: a finite-index d subgroup of a rank n free group has rank dn

Statement

If $H$ has finite index $d$ in a rank- $n$ free group, then $rank (H) = d n$ .

Facts & Assumptions

Given: The false claim above.

[L1]

The correct formula is $rank (H) = 1 + d (n - 1)$ (The Schreier index-rank formula).

Refutation

technique · direct

1.1givenconstruct

Consider the index-two subgroup $H \leq F (a, b)$ consisting of words with even exponent sum in $a$ . Here the ambient free group has rank $n = 2$ and $d = [F (a, b) : H] = 2$ .

2.1L1step 1.1algebra

By [L1], $rank (H) = 1 + 2 (2 - 1) = 3$ , whereas the false formula predicts $d n = 4$ .

3.1step 2.1∎

Since $3 \neq 4$ , this subgroup refutes the statement.

Depends on

The Schreier index-rank formula

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

J. S. Milne, Group Theory, Version 4.01 (standard reference, not scraped)