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0answers
47 views

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 ...
2
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1answer
151 views

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 ...
2
votes
1answer
103 views

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 ...
4
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0answers
154 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 ...
6
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1answer
436 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 = ...
11
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1answer
175 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 ...
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0answers
717 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 ...
4
votes
2answers
217 views

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 ...
9
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1answer
232 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 ...
5
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0answers
150 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. ...
1
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0answers
141 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: ...
0
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0answers
154 views

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 ...
14
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1answer
676 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 ...
9
votes
1answer
445 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 ...
1
vote
1answer
262 views

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 ...
2
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3answers
263 views

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 ...
0
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1answer
117 views

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 ...
0
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1answer
118 views

$p$-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 ...
1
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1answer
207 views

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, ...
2
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1answer
490 views

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 ...
3
votes
1answer
263 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 ...
5
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0answers
307 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 ...
3
votes
2answers
257 views

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}, ...
0
votes
1answer
421 views

Maximal subgroups of a finite p-group

I want to prove the following: Let $G$ be a finite abelian $p$-group that is not cyclic. Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup ...
4
votes
2answers
603 views

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 ...
6
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1answer
937 views

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 ...
3
votes
1answer
481 views

Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one ...
5
votes
0answers
293 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 ...
12
votes
2answers
600 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 ...
0
votes
1answer
366 views

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 ...
26
votes
1answer
1k 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 ...
2
votes
1answer
513 views

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 ...
4
votes
1answer
268 views

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 ...
2
votes
2answers
385 views

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 ...
4
votes
4answers
433 views

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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1answer
387 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 ...
2
votes
1answer
128 views

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 ...
6
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1answer
189 views

Positive cone of a subgroup of Z^n

This question sounds like it should be very well known, but for some reason I failed to find a decent answer anywhere. Let $G\subset\mathbb{Z}^n$ be a subgroup, and $G_+=G\cap\mathbb{Z}_{\ge0}^n$ be a ...
0
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3answers
249 views

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 ...
3
votes
1answer
501 views

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 ...
0
votes
1answer
312 views

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
1
vote
1answer
388 views

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. ...
0
votes
1answer
195 views

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 ...
3
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1answer
919 views

Isomorphic Abelian Group [closed]

How many different non-isomorphic Abelian groups of order n are possible ??
7
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2answers
1k 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
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2answers
600 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 ...
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3answers
2k 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 ...
8
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1answer
1k 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 ...
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3answers
1k 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 ...
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2answers
2k views

Any factor group of a finite abelian group is isomorphic to some subgroup

If you visit this link, you'll see at the top of the PDF view. Basic properties of finite abelian groups: Every quotient group of a finite abelian group is isomorphic to a subgroup. If the above ...