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 labeled Schreier coset graph of a subgroup of a free group

Definition

Let $F (X)$ be a free group on a set $X$ , and let $H \leq F (X)$ . The labeled Schreier coset graph of $H$ in $F (X)$ is the pointed directed graph whose vertices are the right cosets

$V = {H g : g \in F (X)},$

with base vertex $H$ , and whose directed edges are the $x$ -labeled arrows

$H g \overset{x}{\to} H g x (g \in F (X), x \in X) .$

Traversing an $x$ -edge backward is read as $x^{- 1}$ . Hence a path labeled by a word $a_{1} \dots a_{n}$ on $X ⊔ X^{- 1}$ starts at a coset $H g$ and ends at $H g a_{1} \dots a_{n}$ .

Depends on

Used by

Schreier transversals and Schreier systems Definition
A Schreier coset graph and its spanning-tree basis Example
The Schreier coset graph is connected and deterministic Lemma

Dependency tree · two levels

6 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)
C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)