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

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Schreier generators in the right-coset convention

Definition

Let $F (X)$ be a free group, let $H \leq F (X)$ , and let $T$ be a Schreier system of right-coset representatives. For $t \in T$ and $x \in X$ , let $\overline{t x} \in T$ denote the chosen representative of the coset $H t x$ .

The corresponding Schreier generator is

$s (t, x) : = t x {\overline{t x}}^{- 1} .$

This is the right-coset convention: the inverse falls on the representative of the product coset $H t x$ , not on $t$ . Inverse letters are handled in the rewriting map by the rule

$t x^{- 1} {\overline{t x^{- 1}}}^{- 1} = s (\overline{t x^{- 1}}, x)^{- 1},$

so the distinguished generators are indexed only by the basis letters $x \in X$ .

Depends on

Schreier transversals and Schreier systems

Used by

An arbitrary transversal need not give the reduced Schreier basis Counterexample
The Schreier rewriting map Definition
FALSE: the raw Schreier generators are always a free basis False statement
Every Schreier generator lies in the subgroup Lemma
Tree Schreier generators are freely independent Lemma
The Reidemeister-Schreier presentation theorem Theorem

Dependency tree · two levels

3 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)