scholarly journals COMPATIBILITY JSJ DECOMPOSITION OF GRAPHS OF FREE ABELIAN GROUPS

2013 ◽  
Vol 23 (08) ◽  
pp. 1837-1880
Author(s):  
BENJAMIN BEEKER
Keyword(s):  
Abelian Groups ◽  
Finite Graph ◽  
Finitely Generated ◽  
Graph Of Groups ◽  
Jsj Decomposition ◽  
Free Abelian Groups

A group G is a v GBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We describe the compatibility JSJ decomposition over abelian groups. We prove that in general this decomposition is not algorithmically computable.

On autocommutators and the autocommutator subgroup in infinite abelian groups

2018 ◽  
Vol 30 (4) ◽  
pp. 877-885
Author(s):  
Luise-Charlotte Kappe ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Automorphism Group ◽  
Abelian Groups ◽  
Torsion Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Minimal Order ◽  
Autocommutator Subgroup ◽  
Free Abelian Groups ◽  
Torsion Free

Abstract It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

scholarly journals Some cancellation theorems

1980 ◽  
Vol 30 (1) ◽  
pp. 87-97 ◽  
Author(s):  
A. R. Shastri
Keyword(s):  
Bilinear Form ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Equivalence Classes ◽  
Finitely Generated ◽  
Free Abelian Groups ◽  
Cancellation Theorem

AbstractIf G, H and B are groups such that G × B ≃ H × B, G/[G, G]. Z(G) is free abelian and B is finitely generated abelian, then G ≃ H. The equivalence classes of triples (Vξ,A) where Vand A are finitely generated free abelian groups and ξ: V⊗ V → A is a bilinear form constitute a semigroup B undera natural external orthogonal sum. This semigroup B is cancellative. A cancellation theorem for class 2 nilpotent groups is deduced.

scholarly journals Topological realizations and fundamental groups of higher-rank graphs

2015 ◽  
Vol 59 (1) ◽  
pp. 143-168 ◽  
Author(s):  
S. Kaliszewski ◽  
Alex Kumjian ◽  
John Quigg ◽  
Aidan Sims
Keyword(s):  
Fundamental Group ◽  
Abelian Groups ◽  
Fundamental Groups ◽  
Crossed Products ◽  
Finitely Generated ◽  
Topological Realization ◽  
Category Equivalence ◽  
Free Abelian Groups ◽  
Higher Rank ◽  
Projective Limits

AbstractWe investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graphΛ, this functor determines a category equivalence between the category of coverings ofΛand the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions fork-graphs: projective limits and crossed products by finitely generated free abelian groups.

The Residual Finiteness of Polygonal Products—Two Counterexamples

1994 ◽  
Vol 37 (4) ◽  
pp. 433-436 ◽  
Author(s):  
R. B. J. T. Allenby
Keyword(s):  
Abelian Groups ◽  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

scholarly journals Intersection problem for Droms RAAGs

2018 ◽  
Vol 28 (07) ◽  
pp. 1129-1162
Author(s):  
Jordi Delgado ◽  
Enric Ventura ◽  
Alexander Zakharov
Keyword(s):  
Abelian Groups ◽  
Direct Products ◽  
Finitely Generated ◽  
Finite Sets ◽  
Free Products ◽  
Intersection Problem ◽  
Free Abelian Groups ◽  
Defining Graph

We solve the subgroup intersection problem (SIP) for any RAAG [Formula: see text] of Droms type (i.e. with defining graph not containing induced squares or paths of length [Formula: see text]): there is an algorithm which, given finite sets of generators for two subgroups [Formula: see text], decides whether [Formula: see text] is finitely generated or not, and, in the affirmative case, it computes a set of generators for [Formula: see text]. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian groups. We note that most of RAAGs are not Howson, and many (e.g. [Formula: see text]) even have unsolvable SIP.

scholarly journals Torsion-free Abelian groups torsion over their endomorphism rings

1994 ◽  
Vol 50 (2) ◽  
pp. 177-195 ◽  
Author(s):  
Theodore G. Faticoni
Keyword(s):  
Free Group ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Torsion Free Group ◽  
Free Abelian Groups ◽  
Finitely Presented ◽  
Torsion Free

We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End (A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z[X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable. R-module A such that (1) R = End(A), (2) T0 ⊗RA ≠ 0 for each nonzero finitely generated (respectively finitely presented) R-module T0, but (3) T ⊗RA = 0 for some nonzero (respectively nonzero finitely generated). R-module T.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

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