Phill Schultz
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 of G. G contains a direct sum of pure strongly indecomposable groups as a subgroup of finite index.
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Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.
Let $X$ be a compact torsion abelian group. In this paper, we show that an extension of $F_{p}$ by $X$ splits where $F_{p}$ is the p-adic number group and $p$ a prime number. Also, we show that an extension of a torsion-free, non-divisible LCA group by $X$ is not split.
Types in torsion free Abelian groups
该研究聚焦无扭阿贝尔群的逻辑型与等型等价问题，刻画了群内元素及标准二元组的所有可能型，完成了可分无扭阿贝尔群的等型分类。
The torsion ideal lattice of the endomorphism ring of an abelian p-group
本文通过基数不变量系统分类了阿贝尔p-群自同态环的挠理想的格。
In the presented paper we consider some methods of studying of prime ideals in group algebras of abelian groups of finite rank
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.
An indecomposable decomposition of a torsion-free abelian group $G$ of rank $n$ is a decomposition $G=A_1\oplus\cdots\oplus A_t$ where $A_i$ is indecomposable of rank $r_i$ so that $\sum_i r_i=n$ is a partition of $n$. The group $G$ may have decompositions that result in different partitions of $n$. We address the problem of characterising those sets of partitions of $n$ which can arise from indecomposable decompositions of a torsion-free abelian group.
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.
Protori and Torsion-Free Abelian Groups
本文利用紧致阿贝尔群分解定理证明 protorus 的 profinite 子群构成格，引入非阿基米德维数作为分类不变量，并研究其与无挠阿贝尔群的联系。
A proof of the Borel completeness of torsion free abelian groups is presented. This proof differs considerably from the approach of Paolini-Shelah.
Let p be a prime and G be a torsion-free abelian group. A homomorphism from G to the p-adic integers is called a p-adic functional on G. If G has finite rank, then G can be represented as an inductive limit of an inductive sequence of free abelian groups of the same rank, and the group of all p-adic functionals on G is described in terms of this inductive sequence. If this inductive sequence is stationary--i.e., if the homomorphism is the same at every stage--then the group of p-adic functionals is described in particularly simple terms, as a right-submodule that is invariant under the module …
Generalized torsion elements in infinite groups
研究无限群中广义 torsion 元素的性质，证明了有限生成交换陪有限群中广义 torsion 元素的判定条件，并给出了相关群的定量结果和例子。
For any countable torsion subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, any unbounded torsion countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. If $G$ is a bounded torsion countably infinite Abelian group, then it admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. In such a case, a MinAP group topology can be chosen to be complete.
For an Abelian group $G$, any homomorphism $μ\colon G\otimes G\rightarrow G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to addition; the group is called the \textsf{multiplication group} of $G$. Let $\mathcal{A}_0$ be the class of all reduced block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, for groups $G\in \mathcal{A}_0$, we describe groups $\text{Mult}\,G$. We prove that for $G\in \mathcal{A}_0$, the group $\text{Mult}\,…
Some torsion-free solvable groups with few subquotients
本研究构造出满足所有有限生成无挠亚阿贝尔子商均为虚阿贝尔的无限秩有限生成无挠可解群，证明了P. Kropholler的无限秩可解群刻画定理不存在无挠版本。
A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topolo…
Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a …
We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we construct a self-similar group $G=BwrC_{2}$ where $B$ is free abelian of infinite rank.
Let p be a prime. A p-adic functional on a torsion-free abelian group G is a group homomorphism from G to the p-adic integers. The group of all such p-adic functionals is viewed as a p-adic dual group of G, and is studied from the point of view of functional analysis. An analogue of the Hahn--Banach Theorem is proved; this result shows that there are sufficiently many p-adic functionals to be interesting. There is a natural homomorphism from G to its p-adic double dual, and one main result that is proved is that the image of G in this double dual is dense in an appropriate topology. This is us…
We study some close relationships between the classes of transitive, fully transitive and Krylov transitive torsion-free Abelian groups. In addition, as an application of the achieved assertions, we resolve some oldstanding problems, posed by Krylov-Mikhalev-Tuganbaev in their monograph [17]. Specifically, we answer Problem 44 from there in the affirmative by constructing a Krylov transitive torsion-free Abelian group which is neither fully transitive nor transitive. This extends to the torsion-free case certain similar results in the p-torsion case, obtained by Braun et al. in J. Algebra (201…
We prove that the Borel space of torsion-free Abelian groups with domain $ω$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.
We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsion part is in fact determined by the set of all growth functions of a finitely generated Abelian group.
Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We determine, in terms of the structure of G, the kernel of the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, in the case when S is abelian. This provides a classification of all torsion endotrivial kG-modules in that case.
We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group $U$ such that every countable abelian p-group purely embeds in $U$. This is the last result needed to provide a complete solution to Problem 5.1 of [Fuc15] below $\aleph_ω$. We introduce $\aleph_0$-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is $\aleph_0$-strongly homogeneous, and completely characterize the countable $\aleph_0$-strongly homogeneous p-groups.
It has been claimed by Halmos in [Comment on the real line, Bull. Amer. Math. Soc., 50 (1944), 877-878] that if G is a Hausdorff locally compact topological abelian group and if the character group of G is torsion free then G is divisible. We prove that such claim is false, by presenting a family of counterexamples. While other counterexamples are known (see [D. L. Armacost, The structure of locally compact abelian groups, 1981]), we also present a family of stronger counterexamples, showing that even if one assumes that the character group of G is both torsion free and divisible, it does not …
We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H where H is a direct sum (possibly infinite) of cyclic groups. A partial generalization to countable abelian groups of the Gromov rigidity theorem for abelian groups is shown.
Generalized Bassian and other Mixed Abelian Groups with Bounded p-Torsion
本文刻画了具有有界p-挠的混合阿贝尔群，证明所有广义Bassian群具有有限挠自由秩，从而解答了Chekhlov-Danchev-Goldsmith的公开问题，并揭示了其与B+E群的等价及子群关系。
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [4]. Later, this result was extended to all abelian groups [3] and, recently, to all torsion finitely quasihamiltonian groups [7]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [8]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.
We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown's criterion due to Martin Fluch and the author.
On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
本文通过二维方法证明了abelized Johnson核的torsion子群非平凡，并给出了其基数的下界。
Given an aleph_1-free abelian group G we characterize the class C_G of all torsion abelian groups T satisfying Ext(G,T)=0 assuming the continuum hypothesis CH. Moreover, in Godel's constructable universe we prove that this characterizes C_G for arbitrary torsion-free abelian G. It follows that there exist some ugly aleph_1-free abelian groups.
This article extends the works of Gonçalves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups f…
Let $R$ be a commutative local uniserial ring of length $n$, $p$ a generator of the maximal ideal, and $k$ the radical factor field. The pairs $(B,A)$ where $B$ is a finitely generated $R$-module and $A\subset B$ a submodule of $B$ such that $p^mA=0$ form the objects in the category $S_m(R)$. We show that in case $m=2$ the categories $S_m(R)$ are in fact quite similar to each other: If also $R'$ is a commutative local uniserial ring of length $n$ and with radical factor field $k$, then the categories $S_2(R)/\mathcal N_R$ and $S_2(R')/\mathcal N_{R'}$ are equivalent for certain nilpotent categ…
We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.
The Sidon Decomposition Problem in Abelian Groups of Bounded Torsion
本文证明了当对偶群有界挠时，Sidon集是拟独立集的有限并，从而完全解决了Malliavin等人遗留的剩余情形。
Universal co-Extensions of torsion abelian groups
本文证明满足Ext-small的Ab3阿贝尔范畴具有Ab4条件当且仅当每个对象都是Ext-泛的，并刻画了挠阿贝尔群范畴中的co-Ext-泛对象为内射群与各p-分量有界的约化群的直和。
Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups.
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