scholarly journals A note on the Todd-Coxeter coset enumeration algorithm

1976 ◽  
Vol 20 (1) ◽  
pp. 73-79 ◽  
Author(s):  
M. J. Beetham ◽  
C. M. Campbell
Keyword(s):  
Finite Index ◽  
Finitely Generated ◽  
Enumeration Algorithm ◽  
Finitely Presented Group ◽  
Coset Enumeration ◽  
The Given ◽  
Finitely Presented

In (8) Todd and Coxeter described an algorithm for enumerating the cosets of a finitely generated subgroup of finite index in a finitely presented group. Several authors ((1), (2), (5), (6), (7)) have discussed a modification of the algorithm to give also a presentation of the subgroup in terms of the given generators.

Coset Enumeration in a Finitely Presented Semigroup

1978 ◽  
Vol 21 (1) ◽  
pp. 37-46 ◽  
Author(s):  
Andrzej Jura
Keyword(s):  
Finite Number ◽  
Finite Index ◽  
Finite Groups ◽  
Formal Proof ◽  
Finitely Presented Group ◽  
Group B ◽  
Coset Enumeration ◽  
Finitely Presented

The enumeration method for finite groups, the so-called Todd-Coxeter process, has been described in [2], [3]. Leech [4] and Trotter [5] carried out the process of coset enumeration for groups on a computer. However Mendelsohn [1] was the first to present a formal proof of the fact that this process ends after a finite number of steps and that it actually enumerates cosets in a group. Dietze and Schaps [7] used Todd-Coxeter′s method to find all subgroups of a given finite index in a finitely presented group. B. H. Neumann [8] modified Todd-Coxeter′s method to enumerate cosets in a semigroup, giving however no proofs of the effectiveness of this method nor that it actually enumerates cosets in a semigroup.

scholarly journals Some challenging group presentations

1999 ◽  
Vol 67 (2) ◽  
pp. 206-213 ◽  
Author(s):  
George Havas ◽  
Derek F. Holt ◽  
P. E. Kenne ◽  
Sarah Rees
Keyword(s):  
Finite Groups ◽  
Infinite Groups ◽  
Derived Length ◽  
Finitely Presented Group ◽  
Group Presentations ◽  
Coset Enumeration ◽  
Finitely Presented

AbstractWe study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

Determining Subgroups of a Given Finite Index in a Finitely Presented Group

1974 ◽  
Vol 26 (4) ◽  
pp. 769-782 ◽  
Author(s):  
Anke Dietze ◽  
Mary Schaps
Keyword(s):  
Finite Index ◽  
Finite Groups ◽  
Infinite Groups ◽  
Matrix Groups ◽  
Finitely Presented Group ◽  
Basic Object ◽  
Finite Set ◽  
Object Of Study ◽  
Finitely Presented

The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.

scholarly journals On semigroup presentations

1993 ◽  
Vol 36 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Edmund F. Robertson ◽  
Yusuf Ünlü
Keyword(s):  
Group Presentation ◽  
Enumeration Algorithm ◽  
Long Period ◽  
Coset Enumeration ◽  
Finitely Presented

Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd–Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978.In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program.

scholarly journals APPLICATIONS OF p-DEFICIENCY AND p-LARGENESS

2011 ◽  
Vol 21 (04) ◽  
pp. 547-574 ◽  
Author(s):  
J. O. BUTTON ◽  
A. THILLAISUNDARAM
Keyword(s):  
Finite Index ◽  
Coxeter Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
P Deficiency ◽  
Finite Presentation ◽  
Finite Index Subgroup ◽  
Finitely Presented

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large, which implies that Schlage-Puchta's infinite finitely generated p-groups are not finitely presented. We also show that for all primes p at least 7, any group having a presentation of p-deficiency greater than 1 is Golod–Shafarevich, and has a finite index subgroup which is Golod–Shafarevich for the remaining primes. We also generalize a result of Grigorchuk on Coxeter groups to odd primes.

Whitehead Graphs and the Σ1-Invariants of Infinite Groups

1998 ◽  
Vol 08 (01) ◽  
pp. 23-34 ◽  
Author(s):  
Susan Garner Garille ◽  
John Meier
Keyword(s):  
Local Structure ◽  
Universal Cover ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Group ◽  
Finite Presentation ◽  
Finitely Presented Group ◽  
Finitely Presented

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ1(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ1(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ1(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ1(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

Computing with knot quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850074
Author(s):  
Graham Ellis ◽  
Cédric Fragnaud
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Knot Invariant ◽  
Number Formula ◽  
Finitely Presented

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

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