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 On the Computation of Certain Homotopical Functors

1998 ◽  
Vol 1 ◽  
pp. 25-41 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Integral Homology ◽  
Finitely Presented Groups ◽  
The Third ◽  
Finite Module ◽  
Nonabelian Tensor ◽  
A Finite Group ◽  
Finitely Presented

AbstractThis paper provides details of a Magma computer program for calculating various homotopy-theoretic functors, defined on finitely presented groups. A copy of the program is included as an Add-On. The program can be used to compute: the nonabelian tensor product of two finite groups, the first homology of a finite group with coefficients in the arbirary finite module, the second integral homology of a finite group relative to its normal subgroup, the third homology of the finite p-group with coefficients in Zp, Baer invariants of a finite group, and the capability and terminality of a finite group. Various other related constructions can also be computed.

scholarly journals Torsion-free word hyperbolic groups are noncommutatively slender

2016 ◽  
Vol 26 (07) ◽  
pp. 1467-1482 ◽  
Author(s):  
Samuel M. Corson
Keyword(s):  
Free Subgroup ◽  
Hyperbolic Groups ◽  
Topological Groups ◽  
Free Products ◽  
Finitely Presented Groups ◽  
Word Hyperbolic Groups ◽  
Hawaiian Earring ◽  
Free Word ◽  
Finitely Presented ◽  
Torsion Free

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

scholarly journals ON GENERALIZED KNOT GROUPS

2008 ◽  
Vol 17 (03) ◽  
pp. 263-272 ◽  
Author(s):  
XIAO-SONG LIN ◽  
SAM NELSON
Keyword(s):  
Finite Group ◽  
Natural Number ◽  
Conjugacy Classes ◽  
Finitely Presented Groups ◽  
Finitely Presented

Generalized knot groups Gn(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G1(K) in this series of finitely presented groups. For each natural number n, G1(K) is a subgroup of Gn(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism G1(SK) ≅ G1(GK). From the presentations of Gn(SK) and Gn(GK), for n > 1, it seems unlikely that Gn(SK) and Gn(GK) would be isomorphic to each other. Curiously, we are able to show that for any finite group H, the numbers of homomorphisms from Gn(SK) and Gn(GK) to H, respectively, are the same. Moreover, the numbers of conjugacy classes of homomorphisms from Gn(SK) and Gn(GK) to H, respectively, are also the same. It remains a challenge to us to show, as we would like to conjecture, that Gn(SK) and Gn(GK) are not isomorphic to each other for all n > 1.

Subgroups of finitely presented groups

1961 ◽  
Vol 262 (1311) ◽  
pp. 455-475 ◽  
Keyword(s):  
Finite Group ◽  
Genetic Characterization ◽  
Recursively Enumerable Set ◽  
Locally Finite ◽  
Finitely Presented Groups ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented

The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

scholarly journals Density of Metric Small Cancellation in Finitely Presented Groups

2020 ◽  
Vol volume 12, issue 2 ◽  
Author(s):  
Alex Bishop ◽  
Michal Ferov
Keyword(s):  
Experimental Data ◽  
Closed Form ◽  
Small Group ◽  
Upper Bounds ◽  
Small Cancellation ◽  
Lower And Upper Bounds ◽  
Finitely Presented Groups ◽  
Group Presentations ◽  
Interesting Class ◽  
Finitely Presented

Small cancellation groups form an interesting class with many desirable properties. It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no information about "small" group presentations. In this note, we give closed-form formulas for both lower and upper bounds on the density of small cancellation presentations, and compare our results with experimental data. Comment: 18 pages, 12 figures

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′.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

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