Questions tagged [abelian-groups]
For questions about groups whose elements commute.
253 questions
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Abelian group of finite rank
Let given torsion free abelian group $A$ of finite rank. Let for prime number $p$, given that $\cap_i p^iA =\{0\}$. Is it true that for any $p$- torsion abelian group $B$, $\text{Hom}_{\mathbb{Z}}(A, ...
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Looking for a modern source about Ulm Invariants
I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...
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Is the class of additive groups of rings axiomatizable?
I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...
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Counting elements with certain word length in abelian groups
Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
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When does a cogenerator determine a variety?
Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...
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Irreducible characters of finite abelian groups
Let $G$ be finite abelian group and $K$ a field such that $char(K)$ does not divide the order $r$ of $G$. For each divisor $d$ of $r$ let $\omega_d$ be a primitive $d$-root of unity and $a_d:=\frac{\...
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Are the p-adics a direct summand of the direct product of the groups $\mathbb{Z}/p^n\mathbb{Z}$?
The p-adic integers $\mathbb{Z}_p$ can be thought of as a subgroup of the direct product group $P = \prod_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z}$. Are they a direct summand of this group? That is, is ...
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An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$
In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...
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Categorical proof subgroups of free groups are free?
This is a crossport of this question from MSE.
Is there a categorical proof that subgroups of free groups are free?
How about the result that subgroups of free abelian groups are free abelian?
What ...
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Freeness of torsion-free abelian groups
Let $A$ be a countable torsion-free abelian group. The following conditions are well known to be equivalent:
$A$ is free abelian,
every finite rank pure subgroup of $A$ is free abelian.
Consider the ...
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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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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 ...
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The special subgroups of a finite abelian group of rank two
Let $G=\langle a_{1}\rangle\times\langle a_{2}\rangle$ such that $|a_{i}|=2^{k_{i}}$ and $k_{1}>k_{2}$ and $H$ be a subgroup of $G$ that there exists an automorphism of $G$ such that fix only ...
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C* algebras of Almost Periodic Functions
Suppose we take, for example, the $C^*$-algebra which is the sup norm closure of the exponentials $e^{2 \pi i ax}$ where $a \in \mathbb{Z} + \theta \mathbb{Z}$ for $\theta$ an irrational number. This ...
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Is the annihilator of the intersection of two subgroups of a (countable) discrete abelian group generated by the annihilators of the two subgroups?
Let $G$ be a (countable) discrete abelian group and denote by $\hat{G}$ its Pontryagin dual, i.e. the compact abelian group of group homomorphisms $\chi:G \longrightarrow \mathbb{T}$. Recall that, for ...
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Nearly slender abelian groups
Let $\prod_{n=1}^{\infty}\mathbb{Z}$ be the Baer-Specker group (infinite direct product of the additive group of integers) and $\bigoplus_{n=1}^{\infty}\mathbb{Z}$ be the natural subgroup which is the ...
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The existence of non-trivial homomorphisms $\prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$
Let $\prod_{n=1}^{\infty}\mathbb{Z}$ be the Baer-Specker group and $\bigoplus_{n=1}^{\infty}\mathbb{Z}$ be the natural free abelian subgroup. It is known that if $G$ is a countable abelian group with ...
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Name and references for a "twisted" addition in a ring
This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly ...
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Classification of symtrivial modules over a PID
Let us call a module $M$ over a commutative ring $R$ symtrivial if the symmetry $M \otimes M \to M \otimes M, a \otimes b \mapsto b \otimes a$ equals the identity (the same notion applies to arbitrary ...
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Tensor product of topological abelian groups with the reals
Given an abelian group A, the tensor product $A \otimes R$, where R are the reals, is naturally an R-vector space.
Now suppose that A is a topological abelian group (if necessary, we can assume it to ...
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Completion of abelian topological groups
During some exercises I came upon the following questions that I cannot answer myself. Given a topological group $G$ we can build the completion using cauchy sequences and denote this completion by $\...
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Cardinality of the set of elements of fixed order.
Let us consider the group $G:=\mathbb{Z}_N^a$ (the product of the cyclic group with $N$ elements with itself $a$ times). Suppose we are given a number $m$ that divides $N$.
I would like to know how ...
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Looking for concrete description of a category derived from abelian groups
The category of abelian groups $\mathsf{Ab}$ is the $\mathcal{Ind}$-completion of the full subcategory of finitely presentable abelian groups $\mathsf{Ab}_{fp}$. This is not so special, since the ...
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abelian subgroups
Have the groups "PSL(n,q)" and "PSL(n,q).f ", the same maxiaml abelian subgroups or not?(where "PSL(n,q).f " is the extension of PSL(n,q) by the field automorphism of it) Is there any counterexample ...
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The 2-group of extensions
Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...
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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:
...
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Quotients of Abelian Groups
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/(...
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Subgroups of $\mathbb{Z}^n$
I hope that the following problem isn't actually elementary (at least, for the sake of the fact that I'm posting it here), and I apologize if it is. I did try hard to solve it first.
Let $V$ be a $\...
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Finite / uniquely divisible abelian groups
Is there any counter example for the following statement?
STATEMENT:
Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups.
Assume that $F$ is a finite group, and $Q$ is a ...
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Inductive vs projective limit of sequence of split surjections
Let
$$
A_1\twoheadrightarrow
A_2\twoheadrightarrow
A_3\twoheadrightarrow
A_4\twoheadrightarrow
\cdots
$$
be an inductive sequence of countable abelian groups, the connecting homomorphisms of which are ...
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$p$-primary then divisible?
I asked this via MathSE, but haven't got any responces. Sorry for asking it here. Sorry.
We know that in the context of abelian groups, $p$-groups are called $p$-primary groups. I have a question ...
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Why does tensor product in Ab(V) require colimits in V?
In Tom Leinster's book on operads, he gives Ab(V), the category of abelian groups in a symmetric monoidal category V, as an example of a multicategory that doesn't arise from a monoidal category, ...
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Linear algebra of finite abelian groups
If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
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Cancellation theorem for lattices
By a lattice, we mean a finitely generated, free $\mathbb{Z}$-module together with a symmetric bilinear form. Typical examples are the hyperbolic lattices $U$ and the root lattices $A_{n}, D_{n}, E_{n}...
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minimal divisible group
I am trying to prove this:
If a divisible group $E$ containining $A$ is minimal divisible then $A$ is an essential subgroup of $E$.
Let $ < c > =C, \ C\cap A = 0$. Without loss of generality ...
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Hall polynomial when the subgroup is cyclic?
Does anyone know the formula for a Hall polynomial $g_{u,v}^{\lambda}(p)$ when $v$ is the type of cyclic subgroup (ie. $v=(v_{1})$ ) .
http://en.wikipedia.org/wiki/Hall_algebra
I was hoping this ...
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Mysterious property of $\mathbb{Q}$
Hi,
I am currently working through the paper by Bousfield and Gugenheim on rational homotopy theory, and have come to a point where they show why it is important to work over $\mathbb{Q}$, and not ...
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Do the algebraic integers form a free abelian group?
It is a well-known fact, proved in every introductory textbook on algebraic number theory, that if $K$ is an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, then its ring $\mathcal{O}...
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Minimal generation for finite abelian groups
Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:
1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,
2) With orders that are ...
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Definable subsets of the integers as an abelian subgroup?
Consider the integers as a first-order structure in the language {0,+,-} of abelian groups. I suspect that the collection of definable subsets (without parameters) of this structure is an algebra ...
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A question about the additive group of a finitely generated integral domain
Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I ...
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Reference request: a locally cyclic group is isomorphic to a section of the rational numbers
A group $G$ is locally cyclic if whenever $H \le G$ is a finitely generated subgroup then $H$ is cyclic. If $G$ is a locally cyclic group then $G$ is isomorphic to a quotient of a subgroup of the ...
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Categories with canonical factorizations into products satisfying two particular properties
An old splitting theorem for (Hausdorff) locally compact abelian (LCA) groups says that any LCA group $L$ is isomorphic to a direct product of $\mathbb{R}^n$ and $L_1$, where $L_1$ contains a compact-...
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Locally compact abelian groups
First, some preliminaries:
Define an "LCA group" to be a locally compact Hausdorff abelian topological group.
Define "smooth manifold" in a way that requires Hausdorffness, but not connectedness or ...
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The category of Abelian groups with selected elements
Hi,
In his book (Categories for the working mathematician) MacLane speaks (on page 45) about the category of objects (of $\textbf{Ab}$) under $\mathbb{Z}$ which is the comma category $(\mathbb{Z}\...
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Automorphisms of an infinite direct product of abelian groups
Let $G = \prod_p \mathbb{Z}/p\mathbb{Z}$, where $p$ ranges over all primes, considered as an abelian group. What does $\text{Aut}(G)$ (or even $\text{End}(G)$) look like?
I know that that if we take $...
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abelian p-group not divisible [closed]
why if G is an abelian p-group not divisible then exists an element g in G which is not divisible by p?
thanks
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Quotient of subgroups by center.
Let $H \leq G$. Let $Z_G$ denote the center $[G,G]$ the commutator subgroup. Assume $[G,G] \leq Z_G$ (i.e. nilpotent of class 2). Then $G/Z_G$ is abelian since $Z_G$ contains the commutator subgroup. ...
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Does an abelian group acting on a riemaniann manifold define an othogonal foliation?
This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let $V_p$...
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Isomorphic Abelian Group [closed]
How many different non-isomorphic Abelian groups of order n are possible ??
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