##  [Coset Enumeration](/coset-enumeration-0) 

 Definition

An algorithmic process, notably the Todd–Coxeter method, for listing or counting distinct (left or right) cosets of a subgroup H in an ambient group G given by a presentation; output may be a coset table, index, or permutation action on cosets.

 

 

 

 

 

 





## Principle

Principle

Systematically build a coset table by introducing coset symbols and applying generator relations and subgroup definitions to identify and merge cosets until closure or a contradiction arises; the Todd–Coxeter strategy propagates identifications until no new information appears or termination conditions fail.

 

 

 

 

 





## Demonstration

Demonstration

Using a presentation for a finite group G and subgroup H of finite index, run the Todd–Coxeter procedure to fill a coset table; for example, enumerate cosets of a subgroup of index 6 to produce a table of six coset labels and the induced permutation representation of G on those labels.

 

 

 

 

## Misapplication

Misapplication

Applying coset enumeration blindly to an infinite-index subgroup or to a poorly chosen presentation and interpreting non-termination as failure of existence; non-termination can be a property of the presentation or the algorithm rather than of subgroup finiteness.

 

 

 

 

 





## Consequence

Consequence

When successful, coset enumeration yields the index [G:H], explicit coset representatives, and often a permutation representation of G on the cosets, which aids in subgroup lattice exploration and computational group actions.

 

 

 

 

## Reversal

Reversal

Rather than enumerating cosets of H in G, enumerate the cosets of G in a larger overgroup or enumerate double cosets Hackslash G/K; reversal shifts goal from subgroup index to relations between subgroups or overgroups.

 

 

 

 

 





## Boundary

Boundary

Effective when H has finite index or when the presentation admits termination; computational resources and presentation choices can prevent completion; does not by itself give structural properties of H beyond index and action unless supplemented by further computation.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Coset enumeration versus Reidemeister–Schreier and Schreier graph methods: enumeration constructs explicit coset tables and indices, while Schreier techniques build subgroup presentations—both overlap but answer distinct practical questions.

 

 

 

 

 





## Synthesis

Synthesis

Coset enumeration algorithmically constructs the set of cosets of a subgroup by expanding and identifying coset labels under generator relations; when it terminates for finite index one obtains a coset table, index and action, integrating presentation data into explicit combinatorial information about subgroup placement.