Isomorphism of finitely generated solvable groups is weakly universal

2015 ◽  
Vol 219 (5) ◽  
pp. 1639-1644 ◽  
Author(s):  
Jay Williams
Keyword(s):  
Solvable Groups ◽  
Finitely Generated

On the profinite topology on solvable groups

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Khadijeh Alibabaei
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Hnn Extension ◽  
Profinite Topology

AbstractWe show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

scholarly journals Palindromic Width of Finitely Generated Solvable Groups

2015 ◽  
Vol 43 (11) ◽  
pp. 4809-4824 ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Krishnendu Gongopadhyay
Keyword(s):  
Solvable Groups ◽  
Finitely Generated

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

scholarly journals ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

2010 ◽  
Vol 81 (2) ◽  
pp. 317-328 ◽  
Author(s):  
MARCEL HERZOG ◽  
PATRIZIA LONGOBARDI ◽  
MERCEDE MAJ
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Connected Graph ◽  
Solvable Groups ◽  
Maximal Subgroups ◽  
Finite Solvable Group ◽  
Finitely Generated ◽  
Infinite Case ◽  
Locally Graded Group ◽  
A Finite Group

AbstractLet G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

scholarly journals Packing subgroups in solvable groups

2015 ◽  
Vol 25 (05) ◽  
pp. 917-926
Author(s):  
Pranab Sardar
Keyword(s):  
Solvable Group ◽  
Solvable Groups ◽  
Negative Answer ◽  
The Other ◽  
Hyperbolic Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Relatively Hyperbolic Groups ◽  
Relatively Hyperbolic ◽  
Packing Property

We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska–Wise [Packing subgroups in relatively hyperbolic groups, Geom. Topol. 13 (2009) 1945–1988]. In particular, the same is true for all finitely generated subgroups of metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated metabelian subgroup without the bounded packing property. In this example the subgroup is a retract also. Thus we obtain a negative answer to Problem 2.27 of the above paper. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded packing property.

scholarly journals Model theoretic connected components of finitely generated nilpotent groups

2013 ◽  
Vol 78 (1) ◽  
pp. 245-259 ◽  
Author(s):  
Nathan Bowler ◽  
Cong Chen ◽  
Jakub Gismatullin
Keyword(s):  
Nilpotent Group ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Connected Components ◽  
Connected Component ◽  
Finitely Generated ◽  
Image Position

AbstractWe prove that for a finitely generated infinite nilpotent group G with structure (G, ·, …), the connected component G*0 of a sufficiently saturated extension G* of G exists and equalsWe construct an expansion of ℤ by a predicate (ℤ, +, P) such that the type-connected component is strictly smaller than ℤ*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

Sign in / Sign up

Export Citation Format

Share Document