Abelian Groups Of Finite Rank Research Articles

Solving logistic tasks by parallelizing algorithms of the theory of direct decompositions of torsion-free abelian groups

The paper considers the principles of parallelization at marshalling yards and determines their importance. There are presented the methods for direct decompositions of torsion-free Abelian groups of finite rank. There are applied the concepts of “conditionally linear graphs” depicting railway trains and their characteristics, including “quasi-length”, which characterizes the length of the corresponding “quasi-graph”. There has been formalized a graphical approach to the redistribution of railcars of incoming trains at a marshalling yard for the formation of outgoing trains. The corresponding algorithms have been developed.

New examples of indecomposable torsion-free abelian groups of finite rank and rings on them

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative.

Torsion-Free Abelian Groups of Finite Rank with Marked Bases

We define a one-to-one correspondence between torsion-free Abelian groups of finite rank with marked bases and finite sequences of elements of finitely presented modules over the ring of polyadic numbers. This correspondence is a duality of two categories.

The Stabilized Automorphism Group of a Subshift

AbstractFor a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.

Dehn functions of finitely presented metabelian groups

Abstract In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

A matrix description for torsion free abelian groups of finite rank

A matrix description for torsion free abelian groups of finite rank

Torsion-free abelian groups revisited

Let $G$ be a torsion-free abelian group of finite rank. The orbits of the action of Aut($G$) on the set of maximal independent subsets of $G$ determine the indecomposable decompositions. $G$ contains a direct sum of pure strongly indecomposable groups as a subgroup of finite index.

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

A note on Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups in the context of additive groups of rings of finite rank

In the class of torsion-free abelian groups of finite rank, Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups is reduced to indecomposable groups. Torsion-free abelian groups $A$ of rank two such that every associative ring on $A$ is commutative but there exists a~non-commutative ring with the additive group $A$, are classified. Moreover, several valuable results concerning rings on torsion-free abelian groups of rank two achieved by Beaumont, Wisner, Jackett, Aghdam and Najafizadeh are complemented and their proof are greatly simplified.$

On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces

We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.

Centrally essential torsion-free rings of finite rank

It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful.

Behavior of the Auslander condition with respect to regradings

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen–Macaulay property from the same perspective as that for the Auslander condition. A key step of our approach is to establish homological relations between a graded ring [Formula: see text], its quotient ring modulo the ideal [Formula: see text] and its localization ring with respect to the Ore set [Formula: see text], where [Formula: see text] is a homogeneous regular normal non-invertible element of [Formula: see text].

Upper bounds for the dimension of tori acting on GKM manifolds

The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\mathcal{A}(\Gamma,\alpha,\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\Gamma,\alpha,\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action which preserves the almost complex structure, say $(M^{2m},T^{n})$. Then it is shown that $\mathcal{A}(\Gamma,\alpha,\nabla)$ has rank $\ell(> n)$ if and only if there exists an $(m,\ell)$-type GKM graph $(\Gamma,\widetilde{\alpha},\nabla)$ which is an extension of $(\Gamma,\alpha,\nabla)$. Using this combinatorial necessary and sufficient condition, we prove that the rank of $\mathcal{A}(\Gamma_{M},\alpha_{M},\nabla_{M})$ for the GKM graph $(\Gamma_{M},\alpha_{M},\nabla_{M})$ induced from $(M^{2m},T^{n})$ gives an upper bound for the dimension of a torus which can act on $M^{2m}$ effectively. As one of the applications of this result, we compute the rank associated to $\mathcal{A}(\Gamma,\alpha,\nabla)$ of the complex Grassmannian of 2-planes $G_{2}(\mathbb{C}^{n+2})$ with the natural effective $T^{n+1}$-action, and prove that this action on $G_{2}(\mathbb{C}^{n+2})$ is the maximal effective torus action which preserves the standard complex structure.

Decompositions of torsion-free abelian groups

It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.

Indecomposable p-Local Torsion-Free Groups with Quadratic and Cubic Splitting Fields

Indecomposable torsion-free p-local Abelian groups of finite rank with quadratic and cubic splitting field K are characterized. As a consequence, for groups with quadratic splitting field K it is proved that K-decomposable p-local torsion-free Abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.

A torsion-free abelian group of finite rank exists whose quotient group modulo the square subgroup is not a nil-group

The first example of a finite rank torsion-free abelian group A such that the quotient group of A modulo the square subgroup of A is not a nil-group is indicated (in both cases of associative and general rings). In particular, the answer to the question posed by A.E. Stratton and M.C. Webb in [18], Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130, is given for finite rank torsion-free groups. A relationship between nontrivial p-pure subgroups of the additive group of p-adic integers and nontrivial ℤ [p−1]-submodules of the field of p-adic numbers is investigated. In particular, a bijective correspondence between these structures is proven using only elementary methods.

Orbit Dirichlet series and multiset permutations

We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz's $q$-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series' abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture.

Completely decomposable direct summands of torsion-free abelian groups of finite rank

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.

Scott sentences for certain groups

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable $$\varSigma _3$$ Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d- $$\varSigma _2$$ ” (the conjunction of a computable $$\varSigma _2$$ sentence and a computable $$\varPi _2$$ sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of $$\mathbb {Q}$$ . We show that for some of these groups, the computable $$\varSigma _3$$ Scott sentence is best possible, while for others, there is a computable d- $$\varSigma _2$$ Scott sentence.

On state-closed representations of restricted wreath product of groups Gp,d = CpwrCd

Let Gp,d be the restricted wreath product CpwrCd where Cp is a cyclic group of order a prime p and Cd a free abelian group of finite rank d. We study the existence of faithful transitive state-closed (fsc) representations of Gp,d on the rooted m-ary tree for some finite m. The group G2,1, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for d≥2 there are no fsc representations of Gp,d on the p-adic tree. We describe all fsc representations of G=Gp,1 on the p-adic tree obtained via virtual endomorphisms, where the first level stabilizer of the image of G contains its commutator subgroup. Furthermore, for d≥2, we construct fsc representations of Gp,d on the p2-adic tree and exhibit concretely the representation of G2,2 on the 4-tree as a finite-state automaton group.