##  [Reidemeister–Schreier Method](/reidemeister-schreier-method-0) 

 Definition

A constructive procedure to obtain a presentation (generators and relations) for a subgroup H of a group G given a presentation of G; using a Schreier transversal one computes Schreier generators for H and derives relations by rewriting the relators of G in terms of these generators.

 

 

 

 

 

 





## Principle

Principle

Lift coset representatives to express subgroup elements as words in the original generators, produce Schreier generators from transitions between cosets, and translate the ambient relators via rewriting rules to produce relators for the subgroup.

 

 

 

 

 





## Demonstration

Demonstration

Given a finitely presented group G = and a subgroup H of finite index, choose a Schreier transversal of H in G, compute the Schreier generators t_s,g = s g (rep(sg))^{-1} for s in S and coset representatives g, and rewrite each relator r ∈ R to obtain a finite set of relators for H; for example, computing a presentation of an index-2 subgroup of a free group yields a free group of higher rank determined explicitly by Schreier's formula.

 

 

 

 

## Misapplication

Misapplication

Applying the method without ensuring a finite index or neglecting the bookkeeping of coset representatives can lead to infinite or redundant sets of generators and relations; naively simplifying relators may drop necessary relations and give an incorrect presentation.

 

 

 

 

 





## Consequence

Consequence

When applicable, the method produces an explicit (often finite) presentation of H, proving that finite-index subgroups of finitely presented groups are finitely presented and enabling explicit computations in combinatorial and computational group theory.

 

 

 

 

## Reversal

Reversal

The inverse problem—reconstructing the presentation of G from a presentation of H plus coset action data—is nonunique and typically more complicated; subgroup presentations are not canonical and depend on choices of transversal and generators.

 

 

 

 

 





## Boundary

Boundary

Requires a presentation of G and a description of H (often via coset representatives); algorithmic finiteness statements usually assume finite index of H and finiteness of G's presentation — the method becomes cumbersome or non-terminating for infinite-index subgroups or poorly specified transversals.

 

 

 

 

 





## Semantic Tension

Semantic Tension

The method's name is sometimes confused with Reidemeister moves in knot theory; the tension is terminological rather than mathematical—both Reidemeister and Schreier are authors in different combinatorial contexts, so clarity about subgroup presentation is needed.

 

 

 

 

 





## Synthesis

Synthesis

The Reidemeister–Schreier Method systematically converts coset-action data into explicit subgroup generators and relations by constructing Schreier generators from a transversal and rewriting ambient relators, yielding concrete presentations that make subgroup structure computable.