A Note on Root Decision Problems in Groups

1973 ◽  
Vol 25 (4) ◽  
pp. 702-705 ◽  
Author(s):  
Seymour Lipschutz ◽  
Martin Lipschutz
Keyword(s):  
Positive Integer ◽  
Decision Problems ◽  
Finitely Presented Group ◽  
The Relationship ◽  
Finitely Presented

Consider a positive integer r > 1. We say that the rth root problem is solvable for a group G if we can decide for any W ॉ G whether or not W has an rth root, i.e. whether or not there exists V ॉ G such that W = Vr.Baumslag, Boone and Neumann [1] proved that there exists a finitely presented group with all root problems unsolvable. Here we are concerned with the relationship between the different root problems.

Unsolvable Problems in Groups With Solvable Word Problem

1970 ◽  
Vol 22 (4) ◽  
pp. 836-838 ◽  
Author(s):  
James McCool
Keyword(s):  
Word Problem ◽  
Conjugacy Problem ◽  
The Other ◽  
Decision Problems ◽  
Finitely Presented Group ◽  
Other Hand ◽  
Solvable Word Problem ◽  
Finitely Presented ◽  
Effective Procedures ◽  
Solvable Word

Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

DECISION PROBLEMS FOR FINITELY PRESENTED AND ONE-RELATION SEMIGROUPS AND MONOIDS

2009 ◽  
Vol 19 (06) ◽  
pp. 747-770 ◽  
Author(s):  
ALAN J. CAIN ◽  
VICTOR MALTCEV
Keyword(s):  
Markov Property ◽  
Decision Problems ◽  
Open Problems ◽  
Finitely Presented

For some distinguished properties of semigroups, such as having idempotents, regularity, hopficity, etc., we study the following: whether that property or its negation is a Markov property, whether it is decidable for finitely presented semigroups and for one-relation semigroups and monoids. All the results and open problems are summarized in a table.

scholarly journals UNARY FA-PRESENTABLE SEMIGROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250038 ◽  
Author(s):  
ALAN J. CAIN ◽  
NIK RUŠKUC ◽  
RICHARD M. THOMAS
Keyword(s):  
Difficult Problem ◽  
Decision Problems ◽  
Finite Model ◽  
Locally Finite ◽  
Letter Alphabet ◽  
Significant Interest ◽  
Automatic Presentations ◽  
The Relationship ◽  
Completely Simple

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classification of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a difficult problem in general. A restricted problem, also of significant interest, is to ask this question for unary automatic presentations: automatic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally finite, but non-finitely generated unary FA-presentable semigroups may be infinite. Every unary FA-presentable semigroup satisfies some Burnside identity. We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classification is given of the unary FA-presentable completely simple semigroups.

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 Taming the hydra: The word problem and extreme integer compression

2018 ◽  
Vol 28 (07) ◽  
pp. 1299-1381
Author(s):  
W. Dison ◽  
E. Einstein ◽  
T. R. Riley
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Complexity Measure ◽  
Dehn Function ◽  
Dehn Functions ◽  
Fast Growing ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Direct Attack ◽  
Finitely Presented

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a “hydra phenomenon” gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here, we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.

scholarly journals Coherency and Constructions for Monoids

2020 ◽  
Vol 71 (4) ◽  
pp. 1461-1488
Author(s):  
Yang Dandan ◽  
Victoria Gould ◽  
Miklós Hartmann ◽  
Nik Ruškuc ◽  
Rida-E Zenab
Keyword(s):  
Finiteness Condition ◽  
Direct Products ◽  
Finitely Generated ◽  
Rees Matrix Semigroups ◽  
The Relationship ◽  
Brandt Semigroups ◽  
Finitely Presented ◽  
Matrix Semigroups

Abstract A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

scholarly journals Embeddings into groups with only a few defining relations

1974 ◽  
Vol 18 (1) ◽  
pp. 1-7 ◽  
Author(s):  
W. W. Boone ◽  
D. J. Collins
Keyword(s):  
Word Problem ◽  
Finitely Presented Groups ◽  
Fixed Group ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

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