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.

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 On almost finitely generated nilpotent groups

1996 ◽  
Vol 19 (3) ◽  
pp. 539-544 ◽  
Author(s):  
Peter Hilton ◽  
Robert Militello
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Local Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Finitely Generated

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

scholarly journals Automorphisms of nilpotent groups of class two with small rank

1985 ◽  
Vol 39 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Direct Sums ◽  
Matrix Computations ◽  
Central Factor ◽  
Rank One ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractRational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations.In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

scholarly journals Elementary equivalence for finitely generated nilpotent groups and multilinear maps

1998 ◽  
Vol 58 (3) ◽  
pp. 479-493 ◽  
Author(s):  
Francis Oger
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Linear Map ◽  
Multilinear Maps

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.In the proof, we use some properties of commutative rings associated to multilinear maps.

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.

On the Residual Finiteness of Certain Polygonal Products

1989 ◽  
Vol 32 (1) ◽  
pp. 11-17 ◽  
Author(s):  
R. B. J. T. Allenby ◽  
C. Y. Tang
Keyword(s):  
Abelian Groups ◽  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Residually Finite ◽  
Products Of Groups ◽  
Generalized Free Products

AbstractWe give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R, and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R.

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.

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 Annihilator equivalence of torsion-free abelian groups

1992 ◽  
Vol 52 (1) ◽  
pp. 119-129
Author(s):  
P. Schultz ◽  
C. Vinsonhaler ◽  
W. J. Wickless
Keyword(s):  
Structural Properties ◽  
Equivalence Relation ◽  
Abelian Groups ◽  
Equivalence Classes ◽  
Pure Subgroup ◽  
The Other ◽  
Closure Properties ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe define an equivalence relation on the class of torsion-free abelian groups under which two groups are equivalent ifevery pure subgroup of one has a non-zero image in the other, and each has a non-zero image in every torsion-free factor of the other.We study the closure properties of the equivalence classes, and the structural properties of the class of all equivalence classes. Finally we identify a class of groups which satisfy Krull-Schmidt and Jordan-Hölder properties with respect to the equivalence.

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.

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