DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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

Schreier transversals and Schreier systems

Definition

Let $F (X)$ be a free group, let $H \leq F (X)$ , and let ${Sch}_{X} (H)$ be its Schreier graph.

A Schreier transversal for the right cosets of $H$ is a set $T$ of reduced words on $X ⊔ X^{- 1}$ such that every right coset $H g$ is represented by exactly one word of $T$ . For a word $w$ representing the coset $H w$ , write $\overline{w} \in T$ for this chosen representative.

The transversal $T$ is a Schreier system if every initial segment of every word in $T$ again belongs to $T$ . Equivalently, if $x_{1} \dots x_{n} \in T$ , then each prefix $x_{1} \dots x_{j} \in T$ for $0 \leq j \leq n$ , where the empty prefix is the identity word representing the base coset $H$ .

Depends on

Used by

An arbitrary transversal need not give the reduced Schreier basis Counterexample
Schreier generators in the right-coset convention Definition
The Schreier rewriting map Definition
The kernel of an exponent-sum map in a free group Example
FALSE: the raw Schreier generators are always a free basis False statement
Rooted spanning trees and Schreier systems correspond Lemma
The Reidemeister-Schreier presentation theorem Theorem

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

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