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′.

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

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.

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

1968 ◽  
Vol 33 (2) ◽  
pp. 296-297
Author(s):  
J. C. Shepherdson
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Nauk Sssr ◽  
London Mathematical Society ◽  
National Academy Of Sciences ◽  
Doklady Akademii Nauk Sssr ◽  
Finitely Presented Groups ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

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.

scholarly journals A computer-aided analysis of some finitely presented groups

1992 ◽  
Vol 53 (3) ◽  
pp. 369-376 ◽  
Author(s):  
M. F. Newman ◽  
E. A. O'Brien
Keyword(s):  
Finite Group ◽  
Computer Programs ◽  
Free Products ◽  
Finitely Presented Groups ◽  
Cyclic Groups ◽  
Group Presentations ◽  
Computer Aided Analysis ◽  
Computer Aided ◽  
Finitely Presented

AbstractWe answer some questions which arise from a recent paper of Campbell, Heggie, Robertson and Thomas on one-relator free products of two cyclic groups. In the process we show how publicly accessible computer programs can be used to help answer questions about finite group presentations.

scholarly journals FUNCTIONS ON GROUPS AND COMPUTATIONAL COMPLEXITY

2004 ◽  
Vol 14 (04) ◽  
pp. 409-429 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Computational Complexity ◽  
Symmetric Space ◽  
Word Problem ◽  
Exponential Inequality ◽  
Isoperimetric Function ◽  
Length Functions ◽  
Finitely Presented Groups ◽  
Double Exponential ◽  
Context Free ◽  
Finitely Presented

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd–Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

scholarly journals THE GROUPS OF RICHARD THOMPSON AND COMPLEXITY

2004 ◽  
Vol 14 (05n06) ◽  
pp. 569-626 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Word Problem ◽  
Simple Groups ◽  
Direct Products ◽  
Turing Machines ◽  
Group V ◽  
Dehn Functions ◽  
Finitely Presented Groups ◽  
Distortion Functions ◽  
The 1960S ◽  
Finitely Presented

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We give a faithful representation in the Cuntz C⋆-algebra. For the finitely presented simple group V we show that the word-length and the table size satisfy an n log n relation. We show that the word problem of V belongs to the parallel complexity class AC1 (a subclass of P), whereas the generalized word problem of V is undecidable. We study the distortion functions of V and show that V contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of V, the set of Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

scholarly journals UNDECIDABILITY AND THE DEVELOPABILITY OF PERMUTOIDS AND RIGID PSEUDOGROUPS

2017 ◽  
Vol 5 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
HENRY WILTON
Keyword(s):  
Recent Work ◽  
Permutation Group ◽  
Inverse Semigroups ◽  
Existence Problem ◽  
Finitely Presented Groups ◽  
Finite Permutation Group ◽  
Finite Set ◽  
Finitely Presented

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.

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