A recent form of the Todd-Coxeter algorithm, known as the lookahead algorithm, is described. The time and space requirements for this algorithm are shown experimentally to be usually either equivalent or superior to the Felsch and HaselgroveLeech-Trotter algorithms. Some findings from an experimental study of the behaviour of Todd-Coxeter programs in a variety of situations are given.

** Associate Professor, University of Maryland; National Fellow, Hoover Institution, Stanford University. We are grateful to Ron Gilson, Joe Grundfest, Alan Schwartz, and Eric Talley for helpful comments. Todd Cleary provided excellent research assistance. The support of the John M. Olin Program in Law and Economics at Stanford Law School and The Roberts Center for Law, Business, and Corporate ...

A new recursive procedure of the calculation of partition numbers function W (s, d m) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W (s, d m) and their periods τ (G) for the irreducible Coxeter groups and a list for the first twelve symmetric group S m are presented. A least common multiple lcm(m) of the series of the natural num...

A Reidemeister-Schreier program which yields a presentation of a subgroup H of finite index in a finitely presented group G is described in [3J. The program has two stages; first, Schreier generators and Reidemeister relators for H are computed, then the resulting presentation is simplified by eliminating redundant generators and by using a substring searching technique. The Tietze transformati...

Among the six classes of pro-2-groups of finite and fixed co-classes and trivial Schur Multiplicator which studied by Abdolzadeh and Eick in 2009, there are two classes $$S_5=langle a,bmid [b,a^2]=1, a^2=[b,a]^2, (b^2)^{[b,a]}b^2=1rangle$$ and $$S_6=langle a,t,bmid a^2=b^2,[b,a]^2=1, t^a=t^{-1}[b,a], b^t=abarangle$$that have been conjectured to have deficiency zero presentations. In this paper ...

This is the report of a project with the aim to make a new implementation of the Schreier-Sims algorithm in GAP, specialized for matrix groups. The standard Schreier-Sims algorithm is described in some detail, followed by descriptions of the probabilistic Schreier-Sims algorithm and the Schreier-Todd-Coxeter-Sims algorithm. Then we discuss our implementation and some optimisations, and finally ...

We describe a new extension of the Todd–Coxeter algorithm adapted to computing left Kan extensions. The algorithm is a much simplified version of that introduced by Carmody and Walters (Category Theory, Proceedings of the International Conference Held in Como, Italy, 22–28 July 1990. Springer) in 1991. The simplification allows us to give a straightforward proof of its correctness and terminati...

The aim is to apply string-rewriting methods to compute left Kan extensions, or, equivalently, induced actions of monoids, categories, groups or groupoids. This allows rewriting methods to be applied to a greater range of situations and examples than before. The data for the rewriting is called a Kan extension presentation. The paper has its origins in earlier work by Carmody and Walters who ga...

The Todd-Coxeter coset enumeration algorithm is one of the most powerful tools of computational group theory. It may be viewed as a means of constructing permutation representations of nitely presented groups. In this paper we present an analogous algorithm for directly constructing matrix representations over many elds. In fact the algorithm is more general than this, and can be used to constr...