An introduction to presentations of monoid acts: quotients and subacts

The purpose of this article is to introduce the theory of presentations of monoids acts. We aim to construct “nice” general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M-act A and a subact B of A, on the one hand, we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand, we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has a finite complement in an M-act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.     

Presentations of some 3-transposition groups

We classify all 3-transposition groups which are generated by at most five of their 3-transpositions. Modulo a center these are 27 specific groups plus various quotients of a particular group of order 2(3⁴⁹). For each of the 27 groups at least one presentation is given. We also give presentations for many groups of importance in the recent classification of 3-transposition groups with trivial center. Our presentations include ones for the three sporadic 3-transposition groups of Fischer, each on a 3-transposition generating set of minimal size.     

On two problems related to cancellativity

Two decision problems that are related to the properties of right-cancellativity and left-cancellativity, respectively, of the monoidm _T defined by a presentation (Σ;T), are investigated. It is shown that these problems are undecidable in general. In fact, they remain undecidable, even when they are restricted to presentations involving finite Church-Rosser Thue systems. On the other hand, if only finite presentations involving monadic Church-Rosser Thue systems are considered, then these two problems become decidable in polynomial space.     

Positive presentations of the braid groups and the embedding problem

A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.     

Groups of units of integral group rings of Kleinian type

We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.

     

Presentations for crystallographic complex reflection groups

We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group. The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2]. As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part. We then classify the reflections in the groupsW and the minimal number of them needed to generateW, using the diagrams. Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations. The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft.     

The non-abelian simple groupsG, ¦G¦ <106, minimalC-generator pairs, symmetric presentations

In this paper we describe the results of computations to find symmetric presentations based on minimal conjugate generating pairs for the 32 non-abelian simple groups of order less than one million.     

Space complexity of Abelian groups

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism.      

A note on groups and permutation-partition pairs

ANNALI DELL'UNIVERSITA' DI FERRARA - A link between permutation-partition pairs and finite presentations of groups, leading to a graph-theoretical representation of groups, is presented. A...     

Free products and the word problem

An example is given to show that even in a variety whose free word problem (identity problem) is solvable, the free product of two recursive presentations with solvable word problems may have unsolvable word problem. Under some extra conditions on the syntax of the identities defining the variety however, the free product is shown to preserve solvability of the word problem for recursive presentations. The conditions can be checked mechanically, and common varieties such as semigroups and groups satisfy them, as well as many less familiar varieties. The results are obtained by using rewrite-completion techniques.Presented by Ralph Freese.     

Some new efficient soluble groups

We give deficiency zero presentations for an infinite family of finite two generator groups of soluble length six. All previously known finite soluble groups of deficiency zero had soluble length less than six.     

The Tits alternative for non-spherical Pride groups

Pride groups, or ‘groups given by presentations in whicheach defining relator involves at most two types of generators’,include Coxeter groups, Artin groups, triangles of groups, andVinberg's groups defined by periodic paired relations. We showthat every non-spherical Pride group that is not a triangleof groups satisfies the Tits alternative.     

Construction of finite groups amalgams and geometries. Geometries of the group U4(2)

The paper describes methods of constructing of residually connected and flag-transitive geometries and corresponding presentations cf finite groups. The methods are illustrated on the finite simple group U₄(2). As a result all residually connected and flag-transitive geometries of the group U₄(2) with maximsl subgroups as the stabilizers of geometries' elements (exclsding rank 4 geometries with a trivial Borel subgroup) were described. USing these geometries several "natural" presentations of the group U₄(2) were obtained.     

Simulation methodology — An introduction for queueing theorists

Simulation is a widely used methodology for queueing systems. Its superficial simplicity hides a number of pitfalls which are not all as well known as they should be. In particular simulation experiments need careful design and analysis as well as good presentations of the results. Even the elements of simulation such as the generation of arrival and service times have a chequered history with major problems lying undiscovered for 20 years. On the other hand, good simulation practice can offer much more than is commonly realized.     

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.     

Coherent confluence modulo relations and double groupoids

A coherent presentation of an n-category is a presentation by generators, relations and relations among relations. Confluent and terminating rewriting systems generate coherent presentations, whose relations among relations are defined by confluence diagrams of critical branchings. This article introduces a procedure to compute coherent presentations when the rewrite relations are defined modulo a set of axioms. Our coherence results are formulated using the structure of n-categories enriched in double groupoids, whose horizontal cells represent rewriting paths, vertical cells represent the congruence generated by the axioms and square cells represent coherence cells induced by diagrams of confluence modulo. We illustrate our constructions on rewriting systems modulo commutation relations in commutative monoids, isotopy relations in pivotal monoidal categories, and inverse relations in groups.     

A note on the Fibonacci groups

The so-called Fibonacci groups were introduced by Conway in [1]; a more general class of groups is investigated in [2]. In this article, we consider the derived factor-groups of the groups in the latter class, and use two different presentations of them, first to prove that they are all finite and second to give formulae for their orders.     

Permutation presentations of modules over finite groups

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.