Observations on Coset Enumeration

Todd and Coxeter's method for enumerating cosets of nitely generated subgroups in nitely presented groups (abbreviated by Tc here) is one famous method from combinatorial group theory for studying the subgroup problem. Since preex string rewriting is also an appropriate method to study this problem, preex string rewriting methods have been compared to Tc. We recall and compare two of them brieey, one by Kuhn and Madlener 4] and one by Sims 15]. A new approach using preex string rewriting in free groups is derived from the algebraic method presented by Reinert, Mora and Madlener in 14] which directly emulates Tc. It is extended to free monoids and an algebraic characterization for the \cosets" enumerated in this setting is provided. rewriting, Grr obner bases in monoid and group rings.