One-Relator Groups Having a Finitely Presented Normal Subgroup

1978 ◽  
Vol 69 (2) ◽  
pp. 219 ◽  
Author(s):  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Normal Subgroup ◽  
One Relator Groups ◽  
Finitely Presented

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 Fitting quotients of finitely presented abelian-by-nilpotent groups

2014 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
J. R. J. Groves ◽  
Ralph Strebel
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Presented

Abstract.We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

scholarly journals Metabelian groups: Full-rank presentations, randomness and Diophantine problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Albert Garreta ◽  
Leire Legarreta ◽  
Alexei Miasnikov ◽  
Denis Ovchinnikov
Keyword(s):  
Normal Subgroup ◽  
Full Rank ◽  
Direct Decomposition ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Diophantine Problem ◽  
Diophantine Problems ◽  
Finitely Presented ◽  
Finite Presentations

AbstractWe study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if \lvert R\rvert\geq\lvert A\rvert. We further prove that if \lvert R\rvert\leq\lvert A\rvert-1, then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

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 Finitely presented algebras defined by permutation relations of dihedral type

2016 ◽  
Vol 26 (01) ◽  
pp. 171-202 ◽  
Author(s):  
Ferran Cedó ◽  
Eric Jespers ◽  
Georg Klein
Keyword(s):  
Normal Subgroup ◽  
Normal Form ◽  
Symmetric Group ◽  
Cyclic Group ◽  
Index Formula ◽  
Universal Group ◽  
Product Group ◽  
Unique Product ◽  
Dihedral Type ◽  
Finitely Presented

The class of finitely presented algebras over a field [Formula: see text] with a set of generators [Formula: see text] and defined by homogeneous relations of the form [Formula: see text], where [Formula: see text] runs through a subset [Formula: see text] of the symmetric group [Formula: see text] of degree [Formula: see text], is investigated. Groups [Formula: see text] in which the cyclic group [Formula: see text] is a normal subgroup of index [Formula: see text] are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid [Formula: see text] with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group [Formula: see text] of [Formula: see text] is a unique product group, and it is the central localization of a cancellative subsemigroup of [Formula: see text]. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra [Formula: see text] is semiprimitive.

scholarly journals One-relator groups that are residually of prime power order

1975 ◽  
Vol 19 (4) ◽  
pp. 385-409 ◽  
Author(s):  
D. Gildenhuys
Keyword(s):  
Normal Subgroup ◽  
Prime Power ◽  
Prime Power Order ◽  
Residual Properties ◽  
Residual Nilpotence ◽  
One Step ◽  
One Relator Groups ◽  
The One ◽  
Image Position

If C is a class of groups, we denote by RC the class of groups which are residually in C i.e. G ∈ RC if and only if 1 ≠ g ∈ G implies that there exists a normal subgroup N of G such that g ∈ N and G/N ∈ C. A group G is residually a finite p-group if it belongs RFp, where Fp denotes the class of finite p-groups. One also says that the groups in RFp are residually of order equal to a power of the prime p. Given a group G with one defining relator r, one might ask for conditions on the “form” of the relator that would guarantee that G have certain residual properties. In this context, Baumslag (1971) has proved that if all the exponents of the generators appearing in r are positive, then G is residually solvable. In the same paper he also concerned himself with the residual nilpotence of one-relator groups, and found that the situation there was much more complicated. If one goes one step further and asks for conditions that will ensure that for a given prime p the one-relator group be residually a finite p-group, then very little seems to be known. Of course, if one takes r to be one of the generators: then G is freely generated by the remaining generators, and hence is in RFp for all primes p (Mahec (1949), Lazard (1965), 3.1.4). Our main purpose in this paper is to develop methods of generating examples of one-relator groups that are residually of order equal to a given prime p.

On a Class of One-Relator Groups

1980 ◽  
Vol 32 (2) ◽  
pp. 414-420 ◽  
Author(s):  
A. M. Brunner
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Image Position ◽  
The University ◽  
Finitely Presented

In this paper, we consider the class of groups G(l, m; k) which are defined by the presentationwhere k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.

scholarly journals INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 335-344
Author(s):  
MUSTAFA GÖKHAN BENLI
Keyword(s):  
Normal Subgroup ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Free Semigroups ◽  
Finitely Presented ◽  
Indicable Group

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

The BNSR-invariants of the Stein group 𝐹2,3

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Spahn ◽  
Matthew C. B. Zaremsky
Keyword(s):  
Normal Subgroup ◽  
Piecewise Linear ◽  
Unit Interval ◽  
Finitely Generated ◽  
Discrete Character ◽  
Sigma 1 ◽  
Thompson’S Group ◽  
Finitely Presented

Abstract The Stein group F 2 , 3 F_{2,3} is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form 2 p ⁢ 3 q 2^{p}3^{q} ( p , q ∈ Z p,q\in\mathbb{Z} ) and breakpoints in Z ⁢ [ 1 6 ] \mathbb{Z}[\frac{1}{6}] . This is a natural relative of Thompson’s group 𝐹. In this paper, we compute the Bieri–Neumann–Strebel–Renz (BNSR) invariants Σ m ⁢ ( F 2 , 3 ) \Sigma^{m}(F_{2,3}) of the Stein group for all m ∈ N m\in\mathbb{N} . A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of F 2 , 3 F_{2,3} is of type F ∞ \operatorname{F}_{\infty} . Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map F 2 , 3 → Z F_{2,3}\to\mathbb{Z} is of type F ∞ \operatorname{F}_{\infty} , even though there exist maps F 2 , 3 → Z 2 F_{2,3}\to\mathbb{Z}^{2} whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ ∞ ⁢ ( F 2 , 3 ) \Sigma^{\infty}(F_{2,3}) , but there exist (non-discrete) characters that do not even lie in Σ 1 ⁢ ( F 2 , 3 ) \Sigma^{1}(F_{2,3}) . To the best of our knowledge, F 2 , 3 F_{2,3} is the first group whose BNSR-invariants are known exhibiting these properties.

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