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We study the fundamental group of one–dimensional spaces. Among the results we prove are that the fundamental group of a second countable, connected, locally path connected, one–dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and that the fundamental group of a compact, one–dimensional, connected metric space embeds in an inverse limit of finitely generated free groups.

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. Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions. 1. A General Point of View A natural goal in mathematical theories is a full description of the objects that are investigated. This goal has been successfully achieved in some cases, for example all finite abelian groups and with much more effort all finite simple groups. More often one restricted the research activity firstly to more modest problems like the pure existence of any object with som...

this paper we restrict ourselves to the construction of two-dimensional translation planes by working with certain net replacements in the Desarguesian plane which we call "nests". We discuss several infinite families of translation planes so obtained, including a discussion of their collineation groups and some related geometrical properties, and then survey recent attempts at characterizing these planes in terms of the collineation groups admitted. Some open problems and new directions for research will also be mentioned. 2 PRELIMINARY RESULTS

This paper presents a novel technique of using Prolog with never instantiated variables to manipulate a range of algebraic structures. The paper argues that Prolog is a powerful and underrated tool for use in computational number theory. A detailed example is presented in this extended abstract, and several in the full paper, showing the advantages of using this technique. The detailed example is an application of higher dimensional category theory which has been used for solving problems in this area. 1 Introduction Among the many problems dealt with in computational algebra two important classes of problems deal with enumerating algebraic structures with certain properties and manipulating or calculating with the elements of such structures. For particular algebraic structures very efficient solutions to these problems are known and such solutions are typically made available as parts of one or more of the large computational packages now available (which include for example Cayley ...

this paper we develop these ideas further, in the context of the simple group J 3 . This group was chosen firstly because it has an outer automorphism group of order 2, which introduces extra complications, and secondly because it is reasonably small (of order 50232960) so we can do quite a large number of calculations in the group. Our main aims at this stage are: 1. To pass from J 3 :2 to J 3 and (as far as possible) vice versa

We classify all rm, residually connected coset geometries, on which the group PSL(3; 4) acts as a ag-transitive automorphism group ful lling the residually weakly primitive condition: The stabilizer of any ag F acts primitively on the elements of some type in the residue F . We demand also that every residue of rank two satis es the intersection property. We give geometric constructions for all geometries obtained.

Abstract. We determine all firm and residually connected rank 2 geometries on which a Suzuki simple group Sz(q) acts flag-transitively and residually weakly primitively. 1. Introduction In 1986, Francis Buekenhout advocated in [1] a systematic search of the geometries to be obtained from a group and some of its subgroups.