International Journal of Algebra and Computation

We prove that every finite subgroup of a hyperbolic group G can be conjugated to a 2δ+1 neighborhood of the identity element, where δ is the hyperbolicity constant for G with respect to a given generating set. This gives an upper bound for the size of such finite subgroups in terms of δ and the number of generators for G.

In this paper we describe a share package of functions for computing with finite, permutation crossed modules, cat¹-groups and their morphisms, written using the group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat¹-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module , and the corresponding monoid of sections of a cat¹-group , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of , are used to construct the actor crossed module of which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat¹-structures on groups of order at most 30.

Let W be a finite language and let W^c be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in W^c. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

The purpose of this paper is to show that the lattice of permutations on a n -element set is bounded. This result strengthens the semi-distributive nature of the lattice . To prove this property, we use a characterization of the class of bounded lattices in terms of arrows relations defined on the join-irreducible elements of a lattice or, more precisely, in terms of the A-table of a lattice.

This paper is concerned with the structure of semigroups of implicit operations on various subpseudovarieties V of ∩ , where and are the pseudovarieties of all semigroups S in which each regular -class is, respectively, a rectangular group and a group, and where is the pseudovariety of semigroups locally in As an application, we give a characterization of the variety of languages recognized by semigroups in V and derive some join decompositions of pseudovarieties.