Computation in wordhyperbolic groups invited talk abstract. This paper reports on a facility of the anu nq program for computation of nilpotent groups that satisfy an engeln identity. The subject has attracted interest because for many interesting groups including most of the sporadic groups it is impractical to perform calculations by hand. Aug 14, 2010 read finitely presented mvalgebras with finite automorphism group, journal of logic and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A first issue concerns the construction of unique representations which support efficient implementations of various procedures. Pdf finitely presented expansions of groups, semigroups. Here are html and pdf versions of the manuals of the gap 4. Finitely representable spatial objects and efficient computation. Applications of computational tools for finitely presented groups.
However, we also construct an infinite family of finitely presented finitely annihilated groups with cyclic abelianisation. The details are over my head i am not a group theorist, hardly even a mathematician, but i have it on good hearsay that at one time the existence of a finitely generated infinite simple group was known, but the existence of a finitely presented infinite simple group was still an unsolved problem. Gap uses a special representation for these groups which is created in a different way. This paper discusses issues that relate to the finite representation of database entities that may have an infinite size. Bring the puzzle into a form where each row has the same color. It is a subgroup of the permutation group s 16 which is generated by 9 permutations of order 4. Felsch, spas users reference manual, lehrstuhl d fur mathematik, rwth aachen. Every set of defining relations between the elements of any finite generating set of a finitelypresented group contains a finite set of defining relations in these generators. Finitely presented mvalgebras with finite automorphism group. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute. An efficient data structure for completion of finitely presented groups.
A general computational approach for investigating finitely presented groups by way of quotients and subgroups is described and examples are presented. Another special case are groups given by polycyclic presentations. Research in computational group theory, an active subfield of computational algebra, has emphasized four areas. Computation with finitely presented groups by charles c. Finitely presented groups sage reference manual v9. Notation for sets and functions, basic group theory, the symmetric group, group actions, linear groups, affine groups, projective groups, finite linear groups, abelian groups, sylow theorems and applications, solvable and nilpotent groups, pgroups, a second look, presentations of groups, building new groups from old. It is wellknown that a finitely presented group is word hyperbolic if and only if it has subquadratic and thus linear dehn function. Sl2 representations of finitely presented groups computation with finitely presented groups encyclopedia of mathematics and its applications research in computational group theory, an active subfield of computational algebra, has emphasized four areas. The lll lattice reduction algorithm and various algorithms for hermite and smith normal forms are used to study the abelian quotients of a finitely presented group. When is a finitely generated group finitely presented. Notes on computational group theory, chs iii and iv. Coset diagrams in the study of finitely presented groups with an application to quotients of the modular group. Application of computational tools for finitely presented groups. Dooley, editor, proceedings of the 1999 international symposium on symbolic and algebraic computation.
However a theorem of graham higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group. The formulation of the word, conjugacy, and isomorphism problem for finitely presented f. This paper describes the theory and practice of a method for obtaining information about the finite and abelianbyfinite quotients ofg, which often allows computation about larger quotients of the group than has been possible by more traditional methods. Encoding and detecting properties in finitely presented groups. Computation with finitely presented groups, cup, 1994. It is concerned with designing and analysing algorithms and data structures to compute information about groups. Constructing matrix representations of finitely presented groups. Finitely generated nilpotent group, finitely presented conjugacyseparable group, finitely presented solvable group, group in which every subgroup is finitely presented, group with solvable conjugacy. From this we can deduce that there are up to isomorphism only countably many finitely generated recursively presented groups. A new approach to computation in finitelypresented groups. Finitely generated nilpotent group, finitely presented conjugacyseparable group, finitely presented solvable group, group in which every subgroup is finitely presented, group with solvable conjugacy problem, polycyclic. Working with finitely presented groups in maple maplesoft. The puzzle is much simpler than the rubik cube puzzle.
Using the fact that every finitely presented group is the fundamental group of the total space of a lefschetz fibration, we define an invariant of finitely presented groups. The author emphasizes the connection with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, from computational number theory, and from computational commutative algebra. The other method of representation, via a gener ating set, is not particularly suited to calculations. Computation with finitely presented groups encyclopedia of mathematics and its applications book 48 kindle edition by sims, charles c download it once and read it. Read finitely presented mvalgebras with finite automorphism group, journal of logic and computation on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. We then compute the invariant of some groups and give bounds for certain groups. Computation with finitely presented groups pdf free download. A group on finitely many generators defined by finitely many relations between these generators.
Algorithms for finitely lpresented groups and their. Under suitable circumstances a finitely presented group can be. This was a large collection of group theory routines driven by a simple. There is an important class of finitely presented groups known as automatic groups, in which many computations can be undertaken efficiently. Cambridge core algebra computation with finitely presented groups by charles c. The relevant details of the algorithm are presented together with results on engeln groups for moderate values of n.
Symbolic computation 1991 12, 427438 constructing matrix representations of finitely presented groups s. We begin with a simple extension of a wellknown result for abelian groups. We want to generalise this idea and to make it a property of groups rather than of their presentations. Some amount of work regarding permutation groups has been done already but we do not have a proper way to represent a group. Finitely annihilated groups bulletin of the australian. Computation with finitely presented groups 1994, by charles sims. Computing with abelian sections of finitely presented groups. Computation with finitely presented groups encyclopedia of mathematics and its applications book 48 kindle edition by sims, charles c download it once and read it on your kindle device, pc, phones or tablets. Computation with finitely presented groups encyclopedia of.
We demonstrate the application of the algorithms for finitely lpresented groups in the investigation of a class of selfsimilar groups. Every set of defining relations between the elements of any finite generating set of a finitely presented group contains a finite set of defining relations in these generators. Pdf finitely presented algebraic systems, such as groups and semigroups, are of foundational interest in algebra and computation. The dehn function of a finitely presented group, first introduced by gromov, is a useful invariant that is closely related to the solvability of the groups word problem. In this paper, my main focus has been to prove that in nite l 1 i1 z 2 i is coarsely equivalent to in nite l 1 i1 z 3 i. Up to an isomorphism, there are countably many such groups. Representations of finite groups pdf 75p download book. It follows from the well known theorems on the algorithmic unsolvability of the word problem and related problems that there are no deterministic methods to answer most questions about the structure of finitely presented groups. Nation, free and finitely presented lattice, extended version, an extension of chapter 2 of lattice theory. Important algorithms in computational group theory include. Linton institut fiir experimentelle mathematik, ellernstr. Some computations with finitely presented groups citeseerx.
Free groups are a special case of finitely presented groups, namely finitely presented groups with no relators. Computation is a key paradigm that accompanies us throughout the thesis, whether in plain sight or not. Mathematiclly speaking you play this game in a finitely presented group. To prove this, he gives conditions to say if a wreath product of groups is finitely presented. A finitely presented group is a group with a presentation that has finitely many generators and finitely many relations. Discrete mathematics and its applications boca raton. Charles sims computation with finitely presented groups. Computation of nilpotent engel groups journal of the. Lefschetz fibrations and an invariant of finitely presented. We apply our work to show that the weight of a nonperfect finite group, or a nonperfect finitely generated solvable group, is the same as the weight of its abelianisation. Computation is a key paradigm that accompanies us throughout the thesis, whether in. Therefore the methods used by gap to compute in finitely presented groups may run. It is a selfsimilar group with an intermediate word growth.
Kodi archive and support file community software vintage software apk msdos cdrom software cdrom software library console living room software sites tucows software library shareware cdroms software capsules compilation cdrom images zx spectrum doom level cd. Youve concluded that the surjection is finitely generated, so is finitely presented by definition, and there is no need to invoke 4, because the module playing the role of in 4 is. Feb 12, 2015 the idea is to be able to represent finite and finitely presented groups in sympy so that these can be inherited by more complex group objects that ior someone else might implement after the summers. The idea is to be able to represent finite and finitely presented groups in sympy so that these can be inherited by more complex group objects that ior someone else might implement after the summers. I think the question is very general and hard to answer. Finitely presented expansions of groups, semigroups, and. The kbmag package provides such functionality, see the package manual for further details. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. These three lectures will concern the fourth class, for which we do not in general have a unique or canonical representation of groups elements.
Computation with finitely presented groups by sims, charles c. Finitely generated nilpotent group, finitely presented conjugacyseparable group, finitely presented solvable group, group in which every subgroup is finitely presented, group with solvable conjugacy problem, polycyclic group full list, more info. View the article pdf and any associated supplements and figures for a period of 48 hours. Computation with finitely presented groups charles c. Gregorac we here generalize a result of murasugi for finitely presented groups 2. Finitelypresented group encyclopedia of mathematics. On proofs in finitely presented groups 4 4 pruned enumeration starting with a successful coset enumeration where the total number of cosets used, t, exceeds the subgroup index, i, it is often possible to prune the sequence of t. Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented. This book describes the basic algorithmic ideas behind accepted methods for computing with finitely presented groups. The work of baumslag, cannonito, and miller on computing nonabelian polycyclic quotients is described as a generalization of buchbergers grobner basis methods to right ideals in the integral group ring of a polycyclic group. Representing subgroups of finitely presented groups by quotient. Charles c sims research in computational group theory, an active subfield of computational algebra, has emphasized four areas.
Finitely presented groups in geometry and topology hassane k. Kone reu summer 2007, university of tennessee at knoxville mentors. Equivalently, it satisfies the ascending chain condition on subgroups. A noetherian group also sometimes called slender groups is a group for which every subgroup is finitely generated. Example of noetherian group every subgroup is finitely. Jan 21, 2009 using the fact that every finitely presented group is the fundamental group of the total space of a lefschetz fibration, we define an invariant of finitely presented groups.
The book describes methods for working with elements, subgroups, and quotient groups of a finitely presented group. Computation with finitely presented groups ebook, 1994. However ive seen a paper by baumslag wreath products and finitely presented groups, 1961 showing, as a particular case, that the lamplighter group is not finitely presented. Locally finitely presented categories with no flat objects. Youve concluded that the surjection is finitely generated, so is finitely presented by definition, and there is no need to invoke 4, because the module playing the role of in 4 is, not an arbitrary finitely presented module.
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