Questions tagged [abelian-groups]

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89
votes
2answers
6k views

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
67
votes
9answers
14k views

Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders $...
33
votes
4answers
8k views

When is Aut(G) abelian

let $G$ be a group such that $\mathrm{Aut}(G)$ is abelian. is then $G$ abelian? This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\mathrm{Aut}(G)$ is cyclic, but ...
32
votes
3answers
3k views

Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me. Let $G$ be a finite abelian group. Consider the following expression. $$ \nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H| $$ It ...
30
votes
1answer
2k views

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}...
26
votes
3answers
6k views

Subgroups of a finite abelian group

Let $$G=\mathbb{Z}/p_1^{e_1}\times\cdots\times\mathbb{Z}/p_n^{e_n}$$ be any finite abelian group. What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by ...
26
votes
0answers
712 views

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. ...
23
votes
2answers
1k views

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 ...
20
votes
4answers
1k views

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 ...
19
votes
4answers
1k views

Constructively, is the unit of the “free abelian group” monad on sets injective?

Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit ...
18
votes
0answers
565 views

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 ...
17
votes
1answer
1k views

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 $\...
17
votes
1answer
1k views

Epimorphisms $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ are split

Consider the additive group of integer sequences $\mathbb{Z}^{\mathbb{N}}$. Why does every epimorphism of groups $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}^{\mathbb{N}}$ split? $(\star)$ Actually this ...
16
votes
1answer
993 views

A possible mistake in Walter Rudin, "Fourier analysis on groups"

I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$): Suppose $E$ is a coset in $\Gamma_2$ ...
15
votes
3answers
991 views

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 ...
15
votes
1answer
477 views

Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$

Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)? Notice that $\Bbb Z$ is not cancellable, so $A \oplus \Bbb Z \...
14
votes
1answer
893 views

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 ...
14
votes
1answer
569 views

$\mathbb{Z}$-module structure of the subring generated by an algebraic number

Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
14
votes
0answers
455 views

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 ...
13
votes
3answers
803 views

Zero-sum partition of an abelian group

This is a question I have been asking myself some 5 years ago. I later got bored by lack of progress, but maybe some additive combinatorialists here know further. I'm not claiming it is conceptual or ...
12
votes
1answer
635 views

How nearly abelian are nilpotent groups?

It is not uncommon to read that "nilpotent groups are 'close to abelian'."1,2 Can this sentiment be made precise in the sense of the Turán and Erdős definition of "the probability that two elements of ...
12
votes
1answer
876 views

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 ...
11
votes
2answers
2k views

Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split? If not, is there an example?
11
votes
1answer
2k views

Where can I easily look up / calculate (abelian) group cohomology?

For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In ...
11
votes
1answer
246 views

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 ...
11
votes
1answer
384 views

Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$

Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$ $$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
10
votes
2answers
852 views

Classification of subgroups of finitely generated abelian groups

A finitely generated abelian group $A$ is isomorphic to a direct sum of cyclic groups. I am interested in an extension of this result on couples of abelian groups $(A,B),$ where $B$ is a subgroup of $...
10
votes
4answers
559 views

How many non-isomorphic abelian subgroups of the permutation group $S_n$?

I am interested in how many (pairwise non-isomorphic) subgroups of the permutation group $S_n$ are abelian. ($n \in \mathbb{N}$ arbitrary and possibly big) Are you aware of any references which treat ...
10
votes
1answer
1k views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ? This question was first posted here.
10
votes
0answers
385 views

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....
9
votes
2answers
228 views

Do these properties of a countable abelian group guarantee a Prüfer subgroup?

Suppose $(G,+)$ is a countable abelian group and $p$ is a prime number such that: The subgroup $pG$ has finite index in $G$, and For every $n \in \mathbb{N}$, $G$ contains an element of order $p^n$. ...
9
votes
1answer
745 views

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 ...
9
votes
2answers
474 views

Is there a non-degenerate quadratic form on every finite abelian group?

Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...
9
votes
1answer
763 views

constructive Serre classes

A Serre class (of abelian groups) is a class of abelian groups closed under subgroups, quotients, and extensions. For instance, finitely generated groups and finite groups are both Serre classes. ...
9
votes
1answer
910 views

Direct product decomposition for infinite abelian groups with constrained torsion

Let $g$ be a positive integer, and let $G$ be a commutative group with the following constraint on its torsion subgroup: there is an injection $G[\operatorname{tors}] \hookrightarrow (\mathbb{Q}/\...
9
votes
1answer
381 views

Positivity of power of positive PSD matrices

Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries. Suppose now that, on top of having nonnegative entries, ...
9
votes
0answers
278 views

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 ...
8
votes
3answers
3k views

Why are divisible abelian groups important?

I just quote wikipedia: "Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups." I am asking for detail ...
8
votes
2answers
376 views

On $p$-groups with abelian automorphism group

Let $G$ be a $p$-group of order $p^{n}\geq p^{7}$ and its automorphism group is elementary abelian $p$-group. Then, it is clear that $G$ is nilpotent of class $2$. However, the converse is not true in ...
8
votes
2answers
2k views

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 ...
8
votes
2answers
407 views

Exact sequence of $n$th powers of abelian groups

Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \...
8
votes
1answer
2k views

Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
8
votes
1answer
214 views

Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?

$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$Question 1: What is an example of a sequence $(X_\alpha)_{\alpha<\kappa}$ of ...
8
votes
1answer
862 views

On the existence of a direct summand containing a fixed subgroup

Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups that they generate are in direct sum $\langle g_1 \rangle \oplus \ldots \oplus \langle g_n \rangle$. Is it ...
8
votes
1answer
3k views

On order of subgroups in abelian groups

I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the Monthly. If so, would you be so kind as to let me know about the main ideas in Beal'...
8
votes
2answers
1k views

Modern reference for integral homology of a finitely generated abelian group

I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Alg&...
8
votes
1answer
404 views

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 ...
8
votes
0answers
285 views

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 ...
8
votes
0answers
1k views

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 $\...
8
votes
0answers
875 views

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 ...