Beside simplices, n-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n-cubes are not classified. In this work, we first show that there are no Coxeter n-cubes in H for n ≥ 10. Then, we show that the ideal ones exist only for n = 2 and 3, and provide a classification. The methods used are of combinatorial and algebraic nature, using prope...

Andrews and Curtis have conjectured that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. It can be difficult to determine whether or not the conjecture holds for a particular presentation. We show that the utility PEACE, which produces proofs based on Todd-Coxeter coset enumeration, can produce ...

Akl, S., G. Laborite, M. Leeder and K. Qiu, On doing Todd-Coxeter coset enumeration in parallel, Discrete Applied Mathematics 34 (1991) 27-35. The Todd-Coxeter coset enumeration method is without any doubt the best known and most used method in computational group theory. Surprisingly, however, it seems that no attempt has yet been made at producing a parallel version of it. This paper reports ...

Groups can be studied using methods from diierent elds such as combina-torial group theory or string rewriting. Recently techniques from Grr obner basis theory for free monoid rings (non-commutative polynomial rings) respectively free group rings have been added to the set of methods due to the fact that monoid and group presentations (in terms of string rewriting systems) can be linked to spec...

Given a finite presentation of a group G, proving properties of G can be difficult. Indeed, many questions about finitely presented groups are unsolvable in general. Algorithms exist for answering some questions while for other questions algorithms exist for verifying the truth of positive answers. An important tool in this regard is the Todd-Coxeter coset enumeration procedure. It is possible ...

A base and strong generating set provides an effective computer representation for a permutation group. This representation helps us to calculate the group order, list the group elements, generate random elements, test for group membership and store group elements efficiently. Knowledge of a base and strong generating set essentially reduces these tasks to the calculation of certain orbits. Giv...

The well-known Todd-Coxeter algorithm [14], which may be viewed as a means of constructing permutation representations of finitely presented groups, remains a primary reference for coset enumeration programs. All the strategies and variants of this algorithm perform essentially the same calculations as the original algorithm, merely choosing different orders in which to process the available in...

The Todd-Coxeter algorithm described in [13] remains a primary reference for coset enumeration programs. It may be viewed as a means of constructing permutation representations of finitely presented groups. A number of effective computer programs for singlecoset enumeration have been described, see, for example, [2, 7, 8]. Enumerating double cosets, rather than single cosets, gives substantial ...

A local acceleration method for primal-dual potential-reduction algorithms is introduced. The method developed here uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function, and can be regarded as a primal-dual variant of the Iri-Imai algorithm based on the multiplicative analogue of Karmarkar's potential function. When iterates are close to an optimal solu...