Direct Sum Of Cyclic Groups Research Articles

The separating Noether number of the direct sum of several copies of a cyclic group

The exact value of the separating Noether number of some finite abelian groups is determined, including the direct sums of cyclic groups of the same order.

On Dedekind domains whose class groups are direct sums of cyclic groups

For a given family (Gi)i∈N of finitely generated abelian groups, we construct a Dedekind domain D having the following properties.(1)Pic(D)≅⨁i∈NGi.(2)For each i∈N, there exists a submonoid Si⊆D• with Pic(DSi)≅Gi.(3)Each class of Pic(D) and of all Pic(DSi) contains infinitely many prime ideals. Furthermore, we study orders as well as sets of lengths in the Dedekind domain D and in all its localizations DSi.

The sandpile group of a thick cycle graph

The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles . A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph’s edge multiplicities.

Rings on a direct sum of cyclic groups

Rings on a direct sum of cyclic groups

On direct sums of cyclic groups

On direct sums of cyclic groups

On direct sums of cyclic groups with one amalgamated subgroup

On direct sums of cyclic groups with one amalgamated subgroup

Abelian p-groups with minimal full inertia

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.

Lamplighter groups, bireversible automata, and rational series over finite rings

We realize lamplighter groups A\wr \mathbb Z , with A a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize A\wr \mathbb Z as a bireversible automaton group if and only if the 2-Sylow subgroup of A has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.

Abelian Groups With Monomorphisms Invariant With Respect to Epimorphisms

If for any injective endomorphism α and surjective endomorphism β of an abelian group there exists its endomorphism γ such that βα = αγ (respectively, αβ = γα), then we say that the group possesses the R-property (respectively, the L-property). We show that if a reduced torsionfree group possesses the R-property or the L-property, then the endomorphism ring of the group is normal. We describe divisible groups and direct sums of cyclic groups possessing the R-property or the L-property.

Organizing symmetry-protected topological phases by layering and symmetry reduction: A minimalist perspective

It is demonstrated that fermionic/bosonic symmetry-protected topological (SPT) phases across different dimensions and symmetry classes can be organized using geometric constructions that increase dimensions and symmetry-forgetting maps that change symmetry groups. Specifically, it is shown that the interacting classifications of SPT phases with and without glide symmetry fit into a short exact sequence, so that the classification with glide is constrained to be a direct sum of cyclic groups of order 2 or 4. Applied to fermionic SPT phases in the Wigner-Dyson class AII, this implies that the complete interacting classification in the presence of glide is ${\mathbb Z}_4{\oplus}{\mathbb Z}_2{\oplus}{\mathbb Z}_2$ in 3 dimensions. In particular, the hourglass-fermion phase recently realized in the band insulator KHgSb must be robust to interactions. Generalizations to spatiotemporal glide symmetries are discussed.

On the endomorphism rings of abelian groups and their Jacobson radical

We give a characterization of those abelian groups which are direct sums of cyclic groups and the Jacobson radicals of their endomorphism rings are closed. A complete characterization of p-groups A for which \(({\textit{EndA}},\mathcal T_L)\) is locally compact, where \(\mathcal T_L\) is the Liebert topology on \({\textit{EndA}}\) is given. We prove that if A is a countable elementary p-group then \({\textit{EndA}}\) has a non-admissible ring topology. To every functorial topology on A a right bounded ring topology on \({\textit{EndA}}\) is attached. By using this topology we construct on \({\textit{EndA}}\) a non-metrizable and non-admissibe ring topology for elementary countable p-groups A.

Free centre-by-(abelian-by-exponent 2) groups

We study free centre-by-(abelian-by-exponent 2) groups. Our main result is a complete description of the centre. It is isomorphic to a direct sum of a free abelian group and a torsion subgroup. The latter is a direct sum of cyclic groups of order two and cyclic groups of order four. We exhibit a generating set consisting of elements of infinite order, order 2, and order 4, such that the centre is the direct sum of cyclic subgroups generated by those generators. Our approach makes essential use of homological methods.

Some Classes of Hopfian Abelian Groups

This paper describes the Hopfian groups in some classes of Abelian groups. A complete description of Hopfian divisible groups is presented; on the basis of this description, the study of Hopficity of arbitrary Abelian groups is reduced to the study of Hopficity of reduced groups. Direct sums of cyclic groups that are Hopfian groups are characterized. Finally, the paper looks at specific examples of rings whose additive groups are Hopfian.

Mahlo cardinals and the torsion product of primary abelian groups

Nunke’s problem asks when the torsion product of two abelian p-groups is isomorphic to a direct sum of cyclic groups. A complete solution to the problem is given using a new invariant, denoted by L G, whose values are certain collections of finite sets of uncountable regular cardinals. This is a refinement of a previous approach to the problem that only worked up to the first cardinal that is weakly Mahlo. The multiplicative properties of L G are then related to the generalized continuum hypothesis.

Similarity and Coincidence Isometries for Modules

AbstractThe groups of (linear) similarity and coincidence isometries of certain modules Γ in d-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide d. In particular, if the dimension d is a prime number p, the factor group is an elementary abelian p-group. This generalizes previous results obtained for lattices to situations relevant in quasicrystallography.

The nice basis rank of a primary abelian group

An abelian p-group G has a nice basis if it is the ascending union of a sequence of nice subgroups, each of which is a direct sum of cyclic groups. It is shown that if G is any group, then G ⊕ D has a nice basis, where D is the divisible hull of pωG. This leads to a consideration of the nice basis rank of G, i.e., the smallest rank of a divisible group D such that G ⊕ D has a nice basis. This concept is used to show that there exist a reduced group G and a non-reduced group H, both without a nice basis, such that G ⊕ H has a nice basis

An Application of Set Theory to ω + n-Totally p ω+n -Projective Primary Abelian Groups

Several equivalent descriptions are given of the class of primary abelian groups whose separable subgroups are all direct sums of cyclic groups; such groups are called ω-totally Σ-cyclic. This establishes the converse of a theorem due to Megibben. For n < ω, this is generalized to a consideration of the class of primary abelian groups whose pω+n-bounded subgroups are all pω+n-projective. The question of whether there are such groups that are proper in the sense that they are neither pω+n-projective nor ω-totally Σ-cyclic is shown to be logically equivalent to a natural question about the structure of valuated vector spaces. Finally, it is shown that both of these statements are independent of ZFC.

A group of non-uniform exponential growth locally isomorphic to $IMG(z^2+i)$

We prove that a sequence of marked three-generated groups isomorphic to the iterated monodromy group of z 2 + i converges to a group of non-uniform exponential growth, which is an extension of the infinite direct sum of cyclic groups of order 4 by a Grigorchuk group.

Nice bases for primary Abelian groups

Suppose that A is an Abelian p-group. It is proved that if p ω A is bounded, then A has a bounded nice basis and if p ω A is a direct sum of cyclic groups, then A has a nice basis. In particular, all Abelian p-groups of length < ω.2 along with all simply presented Abelian p-groups are equipped with bounded nice bases. It is also shown that if length(A)≤ ω.2 and A/p ω A is countable, then A possesses a bounded nice basis as well as if length(A)≤ ω.2 and p ω A is countable, then A possesses a nice basis. Moreover, contrasting with these claims, we demonstrate that if length(A)=ω.2 and A/p ω A is torsion-complete with finite Ulm-Kaplansky invariants, then A does not have a bounded nice basis. If in addition p ω A is torsion-complete, then A does not have a nice basis, respectively. Finally, we construct a summable $p^{\omega+2}$ -projective group (thus a summable group with a nice basis) which is not a direct sum of countable groups. This answers in negative our question posed in (Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 2005). Keywords: Bounded nice basis, Nice basis, Bounded groups, Direct sums of cyclic groups, Summable groups, $p^{\omega+n}$ -projective groups, Simply presented groups, Σ-groups, Torsion-complete groups, Large subgroups, Countable extensions, Bounded extensions Mathematics Subject Classification: 20K10, 20K15

On quasiresolvent periodic abelian groups

This is a continuation of [1]. We introduce the concept of a primarily quasiresolvent periodic abelian group and describe primarily quasiresolvent and 1-quasiresolvent periodic abelian groups. We construct an example of a quasiresolvent but not primarily quasiresolvent periodic abelian group. For a direct sum of primary cyclic groups we obtain criteria for a group to be quasiresolvent, 1-quasiresolvent, and resolvent, and establish relations among them. We construct a set S of primes such that the direct sum of some cyclic groups of orders p ∈ S is not a quasiresolvent group.