# Questions tagged [abelian-groups]

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### The field of fractions of the rational group algebra of a torsion free abelian group

Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions. ...
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### If $A, B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

The question is in the title. If the isomorphism $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely generated this is also ...
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### Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?

Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...
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### Sets which are unions of translates of each other but aren't single translates

I'm a hobbyist mathematician so any question I ask here might be at risk of closure. I hope this one is good enough, but I'm not sure. This is a continuation of two questions I asked on math....
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### An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
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### A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?

Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
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### Examples of uncountable abelian $p$-groups

Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable? By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
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### A conjecture on circular permutations of n elements in an abelian group of odd order

In 2013 I formulated the following conjecture in additive combinatorics. Conjecture. Let $G$ be an additive abelian group of odd order, and let $A$ be a subset of $G$ with $|A|=n>2$. Then, there is ...
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### Hales' generalization of the stacked bases theorem (seeking a proof)

In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
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### What is an example of an integral domain with a module that is 1-separable but not separable?

Let R be an integral domain. All modules under discussion are torsion free unital left R-modules.     An R-module is completely decomposable if it is the direct sum of rank 1 submodules.     An R-...
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### Quantifier elimination of pp-subgroups of modules

This is a model-theoretic questions. Let $M$ be a $R$-module. Our language will be the standard language of modules, i.e. the language of abelian groups together with an unary function symbol for ...
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### Local cross sections in infinite dimensional groups

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples ...
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### Cotorsion-freeness in uncountable products of abelian groups

An abelian group $A$ is cotorsion provided that whenever $A \leq G$ with $G$ abelian and $G/A$ is torsion-free, we have $G \cong A \oplus B$ for some $B \leq G$. An abelian group $A$ is cotorsion-...
Let $A$ be a finitely generated abelian group. Let $c$ be a 2-cocycle on $A$, where $A$ acts trivially on $\mathbb{C}^\times$. It is well-known that the skew-map $$c(a_1,a_2) \longmapsto \frac{c(a_1,... 0answers 86 views ### Who classified varieties that are commutative groups? Who are the authors of the theorems asserting that connected varieties/manifolds which are abelian groups are isomorphic to {\bf R}^k \times {\bf T}^n? In the smooth setting, I guess this is due to ... 0answers 167 views ### Trivial Tate modules Let A be an abelian group, and p a prime. I'll call$$T_p(A) := \text{Hom}_{\mathbf{Z}}(\mathbf{Q}_{p}/\mathbf{Z}_{p}, A).$$If A is finite, then T_p(A) is trivial, but the converse is not ... 0answers 88 views ### A kind of cancellation ; exchange problem for groups For which (m,n,k,l) \in (\mathbb N\cup \{0\})^4 , with m\le n ; k\le l , does there exist a group G with a finite subnormal series with torsion-free Abelian quotients such that G \times \mathbb ... 0answers 65 views ### Are these convex cones polyhedral? I'm actually playing with some convex cones, and I would like to know if there is a chance they would be described by a finite number of inequalities. Let me introduce some notation first. Let n\... 0answers 80 views ### Alternating bihomomorphism is skew of 2-cocycle - relative situation Let G be a finite abelian group. It is well-known that every alternating bihomomorphism \Omega:G\times G \to \mathbb{C}^\times (i.e. \Omega(g,g)=1) arises as the skew \kappa/\kappa^T of a 2-... 0answers 121 views ### Existence of a transfinite sequence of abelian groups having a strange property I am studying a paper which uses the following lemma. The context is irrelevant, as the lemma is only used as a technical trick and has no pointer to a reference or hint in the proof but its link to ... 0answers 61 views ### Pure (ordered) subgroups Let H,G be abelian groups with H \leq G. We say that H is a pure subgroup of G if for every n \in \mathbb N and h \in H the following holds: If h is n-divisible in G, then h is n-... 0answers 68 views ### Symmetric analogue of "alternating bihomomorphism is skew of 2-cocycle" theorem Let G be a finite abelian group. It is well-known that every alternating bihomomorphism \Omega:G\times G \to \mathbb{C}^\times arises as the skew \kappa/\kappa^T of a 2-cocycle \kappa \in Z^2(G,... 0answers 349 views ### Ring of endomorphisms as a criterion of a dimension 1 module Let R be a unital ring and M be an R-module. I have some questions about relation between the ring \operatorname{End}_R M of endomorphisms and the notion of “dimension” of a module. I’m not ... 0answers 240 views ### How many subgroups of order \prod_{1}^{n} p_{i}^{n_{i}} are there in the finite product of cyclic groups? All of the following {p_{i},q_{i}}are prime numbers, {n,m,k} are pre-assigned integers. Consider the product of cyclic groups \prod_{1}^{n} \mathbb Z_{p_{i}^{n_{i}}} then we asked the question: ... 0answers 89 views ### Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients Is there a name for a pair of lattices which have the property given in the title (up to a change of variable)? The following example of a pair captures the property mentioned above:$$(i)\ 1 + 80q^3 ...
Let $G$ be an abelian group and let $A$ and $B$ be subgroups of $G$. Furthermore, let $C$ be a subgroup of $A \cap B$. I would like to find another subgroup $A+B \subseteq D \subseteq G$ so that \$D/(...