Questions tagged [abelian-groups]
For questions about groups whose elements commute.
254 questions
8
votes
2
answers
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 ...
- 22.3k
11
votes
1
answer
3k
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 ...
- 54.6k
2
votes
1
answer
209
views
Constructing an explicit extension of a continuous character on a closed subgroup of a certain locally compact abelian group
Let $ G $ be a locally compact abelian group and $ \omega: G \times G \to \mathbb{T} $ a continuous multiplier on $ G $, i.e.,
\begin{align}
\forall r,s,t \in G: \qquad
\omega(s,t) ~ \omega(r,s + t) &...
- 1,396
2
votes
0
answers
130
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 ...
- 9,111
2
votes
0
answers
96
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 ...
user111524
1
vote
1
answer
145
views
Non-degeneracy of product of group pairings
For $G$ finite abelian group, let $\eta,\omega:G \times G \to \mathbb{C}^\times$ be group pairings. What can I say about the (non-)degeneneracy of the product pairing $\eta \cdot \omega$ in terms of ...
- 1,813
6
votes
2
answers
704
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 ...
- 152
6
votes
1
answer
278
views
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 ...
- 7,203
10
votes
0
answers
428
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....
- 145
2
votes
1
answer
486
views
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{\...
- 581
2
votes
1
answer
205
views
Generalized height of elements in abelian groups
In the book Infinite abelian groups Vol. I by L. Fuchs, on page 154, the notion of the generalized $p$-height of an element in an abelian group is defined, as follows:
Let $A$ be an abelian group ...
- 1,354
1
vote
0
answers
109
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,...
- 1,813
14
votes
0
answers
518
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 ...
- 2,111
9
votes
0
answers
298
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 ...
- 1,169
12
votes
1
answer
311
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 ...
- 1,291
4
votes
4
answers
630
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 ...
- 6,219
2
votes
0
answers
100
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-...
- 1,813
3
votes
2
answers
902
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 ...
- 133
10
votes
1
answer
845
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 ...
- 11.2k
3
votes
1
answer
728
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 to ...
- 2,998
5
votes
0
answers
171
views
Result of Larsen and Lunts on rationality of power series with coefficients in a free abelian group
Let $G$ be a free abelian group (not necessarily finitely-generated) and $F$ be the fraction field of the group ring of $G$. Let $\Theta$ be the set of power series in $F[[t]]$ such that each nonzero ...
- 622
1
vote
1
answer
212
views
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 ...
- 599
2
votes
0
answers
67
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\...
- 1,766
4
votes
1
answer
588
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, ...
- 1,446
8
votes
0
answers
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 $\...
- 373
2
votes
1
answer
159
views
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, ...
- 121
8
votes
1
answer
453
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 ...
- 1,010
4
votes
1
answer
2k
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 ...
- 125
3
votes
1
answer
1k
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 $...
- 3,115
3
votes
1
answer
1k
views
Isomorphic Abelian Group [closed]
How many different non-isomorphic Abelian groups of order n are possible ??
- 149
5
votes
1
answer
432
views
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 ...
- 188
3
votes
0
answers
133
views
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 ...
- 367
2
votes
1
answer
171
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 ...
- 1,291
4
votes
1
answer
382
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 ...
- 63
1
vote
1
answer
68
views
On finite Uniform (Goldie) dimensions
1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions?
2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$...
- 321
1
vote
0
answers
419
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 ...
- 437
4
votes
2
answers
323
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}, E_{n}...
- 41
3
votes
0
answers
46
views
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-...
- 29
8
votes
2
answers
2k
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&...
- 22.2k
1
vote
1
answer
1k
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. ...
- 13
1
vote
1
answer
281
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 ...
- 391
2
votes
1
answer
198
views
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 ...
- 121
2
votes
1
answer
194
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 ...
- 3,513
3
votes
1
answer
767
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 ...
- 133
0
votes
1
answer
564
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 ...
- 233
0
votes
3
answers
295
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 $(\mathbb{Z}\...
- 733
0
votes
1
answer
570
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 ...
- 131
1
vote
0
answers
242
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:
...
- 8,071
0
votes
1
answer
221
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 ...
- 3,184
0
votes
0
answers
197
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 $D/(...
- 205