DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The Schreier rewriting map

Definition

Let $F (X)$ be a free group, let $H \leq F (X)$ , and let $T$ be a Schreier system. For a word

$w = a_{1} \dots a_{n} (a_{j} \in X ⊔ X^{- 1}),$

define successive representatives by

$t_{0} : = 1, t_{j} : = \overline{a_{1} \dots a_{j}} \in T (1 \leq j \leq n) .$

For each $j$ , define the $j$ th rewriting factor $σ_{j}$ by

$σ_{j} : = {\begin{cases} s (t_{j - 1}, x), & a_{j} = x \in X, \\ s (t_{j}, x)^{- 1}, & a_{j} = x^{- 1} \in X^{- 1} . \end{cases}$

The Schreier rewriting map sends $w$ to the word in Schreier generators and their inverses

$τ (w) : = σ_{1} σ_{2} \dots σ_{n} .$

When $w$ represents an element of $H$ , the last coset is $H$ , so $t_{n} = 1$ .

Depends on

Used by

Dependency tree · two levels

5 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

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory (standard reference, not scraped)
M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the Theory of Groups (standard reference, not scraped)