Next: 5. Kuhn and Madlener's Up: Observations on Coset Enumeration Previous: 3. The Subgroup Problem

# 4. The Todd-Coxeter Coset Enumeration Procedure

Todd-Coxeter coset enumeration (TC) is a famous method from combinatorial group theory for studying finitely presented groups. It is based on the following fundamental observations: Presenting a group in terms of generators and relators R corresponds to viewing it as the quotient of the free group (generated by ) by the normal subgroup generated by R. can be viewed as the subgroup of generated by . Notice that if R is finite, , while finitely generated as a normal subgroup of , need not be finitely generated as a subgroup.

Now given a subgroup of generated by a set the index of in is the same as the index of the subgroup generated by in . While it is undecidable whether a subgroup has finite index in a group, TC attempts to verify whether the index is finite.

In the following we will always assume that the group and the subgroup are finitely presented respectively generated, i.e. the sets , R and S are finite. Detailed descriptions of TC procedures can be found e. g. in [11,3,15]. We only list some of the properties and their interpretations here: If the index of in is finite the procedure halts and produces a set of coset representatives and a coset table with entries for each coset c and each . The (unique) coset representative for any word in can be computed by tracing it through the coset table starting with . Moreover, given a total well-founded ordering > on which is additionally compatible with right concatenation, we can associate to each coset c the representative of minimal length4. The coset table gives rise to a convergent prefix string rewriting system as follows: To each coset w, each and the respective coset wa corresponding to , associate a rule5 of the form . This prefix string rewriting system then can be used to determine the coset of a word in by prefix reduction.

Let us illustrate these findings with an example from [3], page 71:

Example 2   Let be the Dyck group D(3,3,2) presented by and , and the subgroup of generated by . The index of in is 4 and TC (using the length lexicographical ordering induced by ) computes the coset representatives and the coset table
 a b a-1 b-1 b b-1 b b-1 b-1 ba-1 b-1 ba-1 b b ba-1 b ba-1 b-1 ba-1
giving rise to the prefix string rewriting system , , , , , , , , , , , , .

The coset representative of the word aba can be deduced by either tracing the coset table: , and , or by prefix reduction: In both cases we find that aba lies in the coset represented by b-1 which is in fact the minimal representative of this coset with respect to the chosen ordering.

Next: 5. Kuhn and Madlener's Up: Observations on Coset Enumeration Previous: 3. The Subgroup Problem
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