LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Every Schreier generator lies in the subgroup

Statement

Let $F (X)$ be a free group, let $H \leq F (X)$ , and let $T$ be a Schreier system. Then every Schreier generator $s (t, x)$ belongs to $H$ .

Facts & Assumptions

Given: A free group $F (X)$ , a subgroup $H \leq F (X)$ , a Schreier system $T$ , and a Schreier generator $s (t, x)$ .

[L1]

By definition, $s (t, x) = t x {\overline{t x}}^{- 1}$ , where $\overline{t x}$ is the chosen representative of the right coset $H t x$ (Schreier generators in the right-coset convention).

Proof

technique · direct

1.1L1given

Because $\overline{t x}$ represents the same right coset as $t x$ , one has $H t x = H \overline{t x}$ .

2.1L1step 1.1∎

Right-multiplying the equality of step 1.1 by ${\overline{t x}}^{- 1}$ gives $H s (t, x) = H$ . Hence $s (t, x) \in H$ .

Depends on

Schreier generators in the right-coset convention

Used by

The nontrivial Schreier generators generate the subgroup Theorem

Dependency tree · two levels

2 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

M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the Theory of Groups (standard reference, not scraped)