Questions tagged [abelian-groups]
For questions about groups whose elements commute.
60 questions with no upvoted or accepted answers
29
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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.
...
- 35.2k
14
votes
0
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518
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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 ...
- 2,111
11
votes
0
answers
430
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Is $(\mathbb{R}, +)$ still injective as long as $(\mathbb{Q},+)$ is?
It is known that the existence of nontrivial injective abelian groups is independent of choice in ZF (or, rather, ZFA). In particular, $\mathbb{Q}$ is not provably injective, much less $\mathbb{R}$, ...
- 433
10
votes
0
answers
428
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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....
- 145
9
votes
0
answers
298
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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 ...
- 1,169
8
votes
0
answers
309
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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 ...
- 3,011
8
votes
0
answers
1k
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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 $\...
- 373
8
votes
0
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1k
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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 ...
7
votes
0
answers
116
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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 ...
- 15.6k
6
votes
0
answers
346
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Uncountable Mittag-Leffler condition?
Let $(X_\alpha)_{\alpha <\kappa}$ be an inverse system of abelian groups.
If $\kappa = \omega$ (or by extension if $\kappa$ is of countable cofinality), then the Mittag-Leffler condition is a ...
- 64k
6
votes
0
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291
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When is every element of a coend of abelian groups contained in one of the summands?
Let $I$ be a small category and let $D : I^{\mathrm{op}} \times I \to \mathsf{Ab}$ be a functor. The coend
$$\int^{i \in I} D(i,i)$$
can be constructed as the direct sum $\bigoplus_{i \in I} D(i,i)$ ...
- 63.1k
6
votes
0
answers
147
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When is $\{s_2-s_1,s_3-s_2,s_1-s_3\}\cap S$ non-empty for any $s_1,s_2,s_3\in S$?
A subset $S$ of an abelian group is a subgroup if and only if it is closed under taking differences; that is, the difference of any two elements of $S$ is in $S$. Suppose, however, that we only know ...
- 23k
5
votes
0
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190
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Can an infinite abelian $p$-group be tall and thin?
Does there exist an abelian $p$-group $A$ with countable Ulm invariants and uncountable height?
Here by height, I mean the minimal ordinal $\rho$ such that $p^\rho A$ is divisible [1]. For an ordinal ...
- 64k
5
votes
0
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202
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Invariant measure on coset space and integrable functions
Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
- 6,180
5
votes
0
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194
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Which rings are the endomorphisms ring of some abelian groups?
Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
- 151
5
votes
0
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444
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Subgroups and quotients of an abelian pro-finite group
It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$.
I'm wondering whether there is a counterpart for profinite groups.
For example is it true ...
- 93
5
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0
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171
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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
4
votes
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212
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When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on stackexchange.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
- 171
4
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63
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Possible questions about the Tate-Shafarevich subgroup of a Galois hypercohomology group?
$\newcommand{\wt}{\widetilde}$
Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$.
There are finite subgroups
$$ {\rm Sha}^1 \subset H^1,\quad ...
- 14.2k
4
votes
0
answers
113
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.2k
4
votes
0
answers
153
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Image of $\rm{lim}^1$ functor
In category of abelian groups, we know that
— values of $\rm{lim}^1$ on countable systems are precisely cotorsion groups
— values of $\rm{lim}^1$ on systems of finitely generated groups are of the ...
- 4,600
3
votes
0
answers
70
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Admissibility of Ulm's invariants
Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define
$$G_{\alpha}=pG_{\beta}.$$
If $\alpha$ is a limit ...
- 31
3
votes
0
answers
173
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Abelian characters and odd perfect numbers?
This question is about applications of abelian characters to odd perfect numbers:
Context and Definitions:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring ...
- 3,104
3
votes
0
answers
107
views
(Non)complete abelian groups in the “transfinite p-adic topology”
For an abelian group $A,$ a prime $p$ and an ordinal $\alpha,$ we recursively define $p^\alpha A$ as a subgroup of $A$ such that $p^0A=A,$
$$p^{\alpha+1}A=p(p^\alpha A) \hspace{5mm} \text{and} \...
- 1,484
3
votes
0
answers
251
views
Commuting real elements in finite groups
Let $p$, $q$, $r$ be three distinct odd primes, and $G$ a finite group with $|G|$ divisible by $p$, $q$, $r$ to the first power only. Let $x,y,z \in G$ be of order $p,q,r$ respectively. Assume
(a.) $[...
- 191
3
votes
0
answers
138
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The group of sequences in $G^{\mathbb{N}}$ that converge to $(0,0,0,\dots)$
Let $G$ be a discrete abelian group and $G^{\mathbb{N}}$ be the direct product (with the product topology), which consists of sequences $(a_1,a_2,a_3,\dots)$ of elements of $G$.
Let $G^{\infty}$ ...
- 517
3
votes
0
answers
327
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Homology $H_{\ast}(T, V)$
Let $A$ be a local domain. We let $T=T(A) $ be the subgroup of $\mathrm{SL}_{2}$ consisting of diagonal matrices and $V$ be the subgroup of unital matrices of $\mathrm{SL}_{2}$; i.e.
$V:=\left\{\left(
...
- 259
3
votes
0
answers
98
views
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 ...
- 2,992
3
votes
0
answers
46
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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-...
- 29
3
votes
0
answers
133
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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 ...
- 367
3
votes
0
answers
61
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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 ...
2
votes
0
answers
63
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Adjoint to "strict twocategory of strict twofunctors"
Let C be the category of strict twofunctors, featuring the addition of a Grothendieck universe. Strict twocategories are categories enriched over the category of categories.
C has an internal hom ...
user531303
2
votes
0
answers
193
views
Decomposition of finite abelian groups of even order if there is an involution
Let $G$ be a finite abelian group and $\sigma :G\rightarrow G$ an automorphism of order two ($\sigma\circ \sigma =id_G$). Denote by $F$ and $A$ the subgroups of fixed and anti-fixed points of $\sigma$ ...
- 813
2
votes
0
answers
408
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Conceptual proof of fundamental theorem of finite abelian groups
I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
- 2,751
2
votes
0
answers
177
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Existence of fully supported element in a finite-dimensional vector space over $\mathbb{F}_p$ (and in finite abelian groups)
Let $V$ be an $n$-dimensional vector space over $\mathbb{F} = \mathbb{Z} / (p)$, the field of $p$ elements, $p$ a prime, with $\{v_1, \dotsc, v_n \}$ a basis for $V$. An element $x \in V$ is called &...
- 191
2
votes
0
answers
70
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Alternating $n$-homomorphism on abelian group is skew of $n$-cocycle
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
$$ c(a_1,a_2) \longmapsto \frac{c(a_1,...
- 1,813
2
votes
0
answers
176
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Trivial Tate modules
Let $A$ be an abelian group, and $p$ a prime.
I'll call $$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 ...
user119470
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
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
2
votes
0
answers
100
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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
2
votes
0
answers
130
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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
1
vote
0
answers
134
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Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
- 11
1
vote
0
answers
64
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Groups with prescribed Ulm invariants
In Kaplansky's book infinite abelian groups he provides (through some exercises) a complete classification of $p^{\infty}$-torsion countable abelian groups in terms of Ulm invariants. In other words ...
- 11
1
vote
0
answers
79
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A general theory of pairings
Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra.
There are also text books for bilinear forms and related quadratic ...
- 231
1
vote
0
answers
125
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Is the commutator of the holomorph of generalized quaternion group abelian?
Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...
- 19
1
vote
0
answers
97
views
A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
- 14.2k
1
vote
0
answers
68
views
Finding a particular kind of basis of subgroup of a lattice generated by non-negative part
For $\mathbf v=(v_1,\ldots,v_n)\in \mathbb Z^n$, let $\operatorname{supp}(\mathbf v):=\{j: v_j \ne 0\}$. For a subset $X$ of $\mathbb Z^n$, define $\operatorname{supp}(X):=\bigcup_{\mathbf v \in X} \...
- 412
1
vote
0
answers
76
views
elementary abelian subgroups with centralizers not connected
Let $G =$ PGL$_{8}(\textbf{C})$. Let $a, b, c, d$ be four representatives of conjugacy classes of involutions in $G$ where $$a = \begin{pmatrix}
-1 & 0\\
0 & I_{7}
\end{pmatrix}, b = \begin{...
- 501
1
vote
0
answers
42
views
Compute 01-vectors in the orbit of a given vector wrt a finitely-generated abelian subgroup of SL(n,ℤ)
Given a vector $v\in\mathbb Z^n$ and pairwise commutative matrices $M_1,\dotsc,M_k\in \operatorname{SL}(n,\mathbb Z)$, how to compute all 01-vectors in the orbit of $v$ with respect to multiplication ...
- 34.4k
1
vote
0
answers
67
views
Eigenvalue function of the representation variety of free abelian group
$\DeclareMathOperator\SL{SL}$Let $\rho:\mathbb Z^n\rightarrow \SL(n,\mathbb C)$ be a representation of a finitely generated free abelian group, by simultaneously triangularization, we can assume that $...
- 502