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 semigroups satisfying x3 = x

1978 ◽  
Vol 84 (1) ◽  
pp. 11-19 ◽  
Author(s):  
J. A. Gerhard
Keyword(s):  
Word Problem ◽  
Upper Bound ◽  
Finitely Generated ◽  
Finitely Generated Groups

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

The History, Classes and Presentations of Groups

2012 ◽  
Vol 430-432 ◽  
pp. 834-837
Author(s):  
Xiao Qiang Guo ◽  
Zheng Jun He
Keyword(s):  
Group Theory ◽  
Algebraic Groups ◽  
Transformation Groups ◽  
Combinatorial Group Theory ◽  
Algebraic Equations ◽  
Historical Sources ◽  
Matrix Groups ◽  
History Of ◽  
History Classes ◽  
Combinatorial Group

First we introduce the history of group theory. Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. Secondly, we give the main classes of groups: permutation groups, matrix groups, transformation groups, abstract groups and topological and algebraic groups. Finally, we give two different presentations of a group: combinatorial group theory and geometric group theory.

On Weighted Geodesics in Groups

1987 ◽  
Vol 30 (1) ◽  
pp. 86-91
Author(s):  
Seymour Lipschutz
Keyword(s):  
Weight Function ◽  
Free Product ◽  
Word Problem ◽  
Minimum Weight ◽  
Minimum Length ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Geodesic Problem ◽  
Solvable Word

AbstractA word W in a group G is a geodesic (weighted geodesic) if W has minimum length (minimum weight with respect to a generator weight function α) among all words equal to W. For finitely generated groups, the word problem is equivalent to the geodesic problem. We prove: (i) There exists a group G with solvable word problem, but unsolvable geodesic problem, (ii) There exists a group G with a solvable weighted geodesic problem with respect to one weight function α1, but unsolvable with respect to a second weight function α2. (iii) The (ordinary) geodesic problem and the free-product geodesic problem are independent.

Sign in / Sign up

Export Citation Format

Share Document