Todd Coxeter AlgorithmEdit
The Todd-Coxeter algorithm is a foundational tool in computational group theory for enumerating the left cosets of a subgroup inside a finitely presented group. Given a presentation G =
The historical impulse behind the Todd-Coxeter algorithm was to provide a systematic procedure for turning questions about abstract, finitely presented groups into concrete, finite combinatorial data. The original ideas were developed by J. A. Todd and H. S. Coxeter in the mid-20th century and have since become standard material in texts on group presentations and coset enumeration. Their work opened a path for many later improvements and refinements that keep the method relevant for today’s computational needs, including implementations in systems such as GAP and Magma (computer algebra system). The algorithm also helped illuminate the relationship between finitely presented groups and their actions on finite sets, a theme that recurs in the study of permutation groups and related structures.
Overview
- Structure of the coset table. The central object is a table with a row for each coset of H in G and a column for each generator x in X (and often for x^{-1} as well). The entry in row i, column x indicates which coset arises when you multiply a representative of coset i by x on the left. The table is built iteratively, creating new coset rows as needed and filling in entries to reflect the action of the generators on the cosets.
- Defining the action. The algorithm attempts to realize G as a group of permutations acting on the set of cosets G/H. In doing so, it enforces the relations R by following words in the generators along the coset paths and ensuring that the resulting product behaves as the identity on the coset set.
- Consistency and completion. A key part of the method is maintaining coherence among the entries: if a path by a word w should bring you back to the starting coset (because w represents the identity in G), the corresponding path in the table must close. When a needed coset is not yet defined, the algorithm introduces a new coset; when conflicting identifications arise, consistency checks are performed, sometimes requiring backtracking or alternative choices. The process continues until all relations are satisfied for all cosets, yielding a complete table when the index is finite.
- Variants and optimizations. Several practical variants exist to improve performance on difficult presentations. The original approach is often augmented by strategies that determine the order in which cosets are extended and how new cosets are created, with methods such as the so-called Felsch approach being widely discussed in the literature. These refinements can dramatically affect running time in practice depending on the structure of the group and the chosen subgroup.
Variants and practical considerations
- Original scheme vs. optimized strategies. The classic Todd-Coxeter framework is frequently described alongside practical optimizations that affect the enumeration order and table growth. The choice of strategy can influence memory usage and speed, especially for groups with large or intricate finite quotients.
- Relationship to other coset techniques. The Todd-Coxeter method sits alongside other approaches in the broader landscape of coset enumeration and permutation-group computation, including algorithmic ideas that share the same goal of producing a concrete action on a finite set when possible. In many software packages, users can select between variants or rely on heuristics that adapt to the input presentation.
- Implementation in modern systems. Today’s computer algebra environments implement the Todd-Coxeter paradigm as part of a larger toolkit for manipulating finitely presented groups, performing tasks such as testing finiteness, constructing permutation representations, and exploring subgroup structure within a finitely presented setting. See for example GAP and Magma (computer algebra system) for practical realizations.
Algorithmic properties and limitations
- Termination and finiteness. The algorithm is guaranteed to terminate with a complete coset table when the subgroup has finite index in a finite group. In general, for infinite groups or infinite index, termination is not guaranteed, and the computation may run indefinitely or require heuristic stopping criteria.
- Computational complexity. The worst-case resource requirements can be substantial, growing with the index [G : H] and the complexity of the relations R. This makes the method powerful for many concrete finitely presented groups but challenging for particularly large or intricate presentations.
- Output interpretation. When the table is complete, the resulting data yields a permutation representation of G on the cosets G/H and gives explicit information about the action of generators on these cosets. This data can be used to derive additional structural information about G and about the subgroup H.
Applications and context
- Finiteness testing and quotient construction. The Todd-Coxeter algorithm is widely used to decide whether a finitely presented group is finite and, if so, to construct its permutation representation and a complete list of cosets.
- Subgroup analysis. By enumerating cosets, one can study indices of subgroups, their generators, and their action on various finite quotients, aiding in the broader study of a group’s lattice of subgroups.
- Computational tools in algebra. The method remains part of standard toolkits in computer algebra systems and is frequently used in research and education to illustrate how abstract presentations translate into concrete permutation actions. See GAP and Magma (computer algebra system) for implementations and practical demonstrations.