The Todd-Coxeter algorithm, discovered by Todd and Coxeter in 1936, is a procedure that can solve the coset enumeration problem. Given a group presentation , where is a set of generators and is a set of relations, and a subgroup , we wish to compute the action of on . The Todd-Coxeter algorithm is guaranteed to succeed in a finite number of steps provided is finite, although the running time is unpredictable. Specifically, no recursive function of and the input can bound the running time.

Note: The rest of the article is under construction and will be finished soon --Superninja 09:51, 12 Aug 2004 (UTC)

One implementation of the algorithm proceeds as follows. Define to be the set of generators and their inverses. Let where the are words of elements of . There are three types of tables that will be used: a coset table, a relation table for each relation in , and a subgroup table. Information is gradually added to these tables, and once they are filled in, all cosets have been enumerated and the algorithm terminates.

The coset table is used to stored the relationships between the known cosets when multiplying by a generator. It has rows representing cosets of and a column for each element of . Let denote the coset of the ith row of the coset table, and let denote generator of the ith column. The entry of the coset table in row i, column j is defined to be (if known) k, where k is such that .