On the algorithmic insolvability of the word problem in group theory

1958 ◽  
pp. 1-122 ◽  
Author(s):  
P. S. Novikov
Keyword(s):  
Group Theory ◽  
Word Problem

scholarly journals A Descriptive View of Combinatorial Group Theory

2011 ◽  
Vol 17 (2) ◽  
pp. 252-264
Author(s):  
Simon Thomas
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Embedding Theorem ◽  
Combinatorial Group Theory ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Combinatorial Group

AbstractIn this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman–Neumann–Neumann Embedding Theorem.

The word problem for one-relator semigroups

1986 ◽  
Vol 99 (1) ◽  
pp. 33-44 ◽  
Author(s):  
James Howie ◽  
Stephen J. Pride
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Semi Group ◽  
Full Generality ◽  
Analogous Question

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.

An Algorithmic Solution for a Word Problem in Group Theory

1964 ◽  
Vol 16 ◽  
pp. 509-516 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Group Theory ◽  
Finite Number ◽  
Word Problem ◽  
Finite Index ◽  
Unit Element ◽  
Systematic Procedure ◽  
Algorithmic Solution ◽  
Finite Set ◽  
The Right

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

scholarly journals Applications of L systems to group theory

2018 ◽  
Vol 28 (02) ◽  
pp. 309-329 ◽  
Author(s):  
Laura Ciobanu ◽  
Murray Elder ◽  
Michal Ferov
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Normal Forms ◽  
Free Groups ◽  
Fundamental Groups ◽  
Context Sensitive ◽  
Rank 2 ◽  
L Systems ◽  
Primitive Sets ◽  
Context Free

L systems generalize context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper, we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the Bridson–Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk’s group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L languages of intermediate growth.

Friends and relatives of BS(1,2)

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Charles F. Miller III
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Teaching Experience ◽  
Geometric Group Theory ◽  
Point Of View ◽  
Fundamental Importance ◽  
Finitely Presented Groups ◽  
Historical Essay ◽  
Central Interest ◽  
Finitely Presented

AbstractAlgorithms, constructions and examples are of central interest in combinatorial and geometric group theory. Teaching experience and, more recently, preparing a historical essay have led me to think the familiar group BS(1,2) is an example of fundamental importance. The purpose of this note is to make a case for this point of view. We recall several interesting constructions and important examples of groups related to BS(1,2), and indicate why certain of these groups played a key role in showing the word problem for finitely presented groups is unsolvable.

The word problem for division rings

1973 ◽  
Vol 38 (3) ◽  
pp. 428-436 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Group Theory ◽  
Recent Work ◽  
Word Problem ◽  
Equivalent Formulation ◽  
Finitely Presented Groups ◽  
First Order ◽  
Division Rings ◽  
Group Presentations ◽  
Finitely Presented ◽  
Horn Sentences

In this paper we prove that the word problem for division rings is recursively unsolvable. Our proof relies on the corresponding result for groups [7], [28], and makes essential use of P. M. Cohn's recent work [11], [13], [15], [16] on division rings.The word problem for groups is usually formulated in terms of group presentations or finitely presented groups, as in [7], [24], [28], [30]. An equivalent formulation, in terms of the universal Horn sentences of group theory, is mentioned in [32]. This formulation makes sense for arbitrary first-order theories, and it is with respect to this formulation that we show that the word problem for division rings has degree 0′.

Sign in / Sign up

Export Citation Format

Share Document